Method and apparatus for identifying a path between source and target states in a space with more than two dimensions

ABSTRACT

Some embodiments of the invention provide a method for identifying a path between a set of source states and a set of target states in a space with more than two dimensions. In some embodiments, the space includes a set of states and a transition map that specifies a set of states that can be reached from each particular state. At least some of the states are non-zero dimensional states. For instance, in some embodiments, the space is a graph that includes points, lines, and surfaces. The method specifies at least one path that starts from one state. It then iteratively specifies new paths by expanding previously specified paths to other states in the space until identifying a path that connects the source and target states. At least one of the expansions of a previously specified path includes an expansion in more than two dimensions of the space.

CLAIM OF BENEFIT TO PROVISIONAL APPLICATION

This patent application claims the benefit of the earlier-field U.S.Provisional Patent Application entitled “Method and Apparatus forRouting”, having Ser. No. 60/396,571, and filed Jul. 15, 2002.

FIELD OF THE INVENTION

The invention is directed towards method and apparatus for identifying apath between source and target states in a space with more than twodimensions.

BACKGROUND OF THE INVENTION

A best-first search is an iterative search that at each iteration triesto extend the partial solution with the best estimated cost. Best-firstsearches have often been used to identify the shortest path betweensource and target points in a multi-point search space. One suchbest-first search is the A* search. To identify a path between sourceand target points, the A* search starts one or more paths from thesource and/or target points. It then iteratively identifies one or morepath expansions about the lowest cost path, until it identifies a paththat connects the source and target points. The typical cost of a pathexpansion in an A* search is an {circumflex over (F)} cost, which is thecost of the path leading up to the path expansion plus an estimated costof reaching a target point from the path expansion.

FIGS. 1–6 provide an example of an A* path search that uses such an{circumflex over (F)} cost. In this example, the search process has tofind the shortest path between a source point 105 and a target point 110in a region 120. The source and target points are part of a multi-pointgrid 115 that is imposed over the region.

As shown in FIG. 2, the search process initially identifies four pathexpansions from the source point 105 to four points 202–208 thatneighbor the source point in the Manhattan directions. In FIGS. 2–6, thesearch process represents each path expansion by using a pathidentifier, called a “drop.” More specifically, the search processrepresents each path expansion from one grid point (a start point) toanother point (a destination point) by (1) specifying a drop, (2)associating the drop with the expansion's destination point, and (3)defining the specified drop's previous drop to be the drop of theexpansion's start point. Drops allow the search to keep track of thepaths that it explores.

The search process specifies four drops 210–216 for the expansions tothe four points 202–208, as illustrated in FIG. 2. It also specifies adrop 218 for the source point 105. The source point drop 218 is theprevious drop of drops 210–216. The source point drop's previous drop isnull, as it is the first drop in the path search.

For each drop 210–216, the search process computes an F cost based onthe following formula:{circumflex over (F)}=G+Ĥ.where (1) G specifies the cost of a path from the source point to thedrop's grid point through the sequence of expansions that led to thedrop, and (2) Ĥ specifies the estimated lower-bound cost from the drop'sgrid point to the target point. When computed in this manner, the{circumflex over (F)} cost of a drop is the estimated cost of thecheapest path that starts at the source point, traverses through thesequence of expansions that led to the drop, and traverses from the dropto the target point.

To simplify the description of the example illustrated in FIGS. 1–6, thedistance between each pair of horizontally or vertically adjacent gridpoints is 1. Accordingly, in FIG. 2, the G cost of each drop 210–216 is1, as the grid point of each of these drops is one grid unit away fromthe source point. In FIGS. 2–6, a drop's Ĥ cost is computed as theManhattan distance between the drop's point and the target point. Hence,the Ĥ cost of drops 210, 212, 214, and 216 are respectively one, three,three, and three, as these distances are respectively the Manhattandistances of the points of these drops from the target point.

After costing these drops, the search process stores the drops 210–216in a priority queue that is sorted based on their {circumflex over (F)}costs. It then retrieves the drop with the lowest {circumflex over (F)}cost from the priority queue. This drop is drop 210. Since this drop'scorresponding point (i.e., point 202) is not the target point, thesearch process then identifies a path expansion from the retrieveddrop's point 202 to point 305. As shown in this figure, this expansionis the only viable expansion from the retrieved drop's point 202 as thesearch has previously reached all other unblocked neighboring gridpoints (i.e., the grid points that are not blocked by an obstacle 315)through less expensive paths. The search process specifies a drop 310for this expansion, and computes this drop's G, Ĥ, and {circumflex over(F)} costs, which are respectively 2, 2, and 4. It then stores thisspecified drop in the priority queue.

After storing drop 310 in the priority queue, the search process mightretrieve either drop 310, drop 212, or drop 214 from the priority queue,as each of these drops has an F cost of 4. However, if the searchprocess retrieved drop 310, it will not expand from this drop to itsneighboring points that are not blocked by obstacle 315 since all theseneighboring points were previously reached less expensively. Also, ifthe search process retrieves drop 214, it will identify drops that willbe more expensive than drop 212.

When the search process retrieves drop 212 from the priority queue, itchecks whether this drop's point is the target point. When it discoversthat it is not, the process (1) identifies expansions to threeneighboring points 402–406 about this drop's point, as shown in FIG. 4,(2) specifies three drops 408–412 for the three identified expansions,as shown in FIG. 4, (3) computes each specified drop's G, Ĥ, and{circumflex over (F)} costs, (4) defines each specified drop's previousdrop (which in this case is drop 212), and (5) stores each newlyspecified drop in the priority queue based on its F cost.

Next, as illustrated respectively in FIGS. 5 and 6, the search processperforms these six operations first for drop 408 and then for drop 510,since these two drops are the ones with the lowest F costs during thenext two iterations of the search process. As illustrated in FIG. 5, thedrop 510 is specified for an expansion about drop 408's point 402.

As shown in FIG. 6, one of the expansions about drop 510 reaches thepoint 110. The search process creates, costs, and stores a drop 615 forthis expansion. It then retrieves this drop in its next iteration, andthen realizes that this drop's point is the target point. Accordingly,at this juncture, it terminates its path search operation. It thencommences a path-embedding, back-trace operation that uses theprevious-drop references of the drops 615, 510, 408, 212, and 218 toidentify the sequence of drops that reached the target point 110 fromthe source point 105. This operation embeds a path along the grid pointsassociated with the identified sequence of drops. FIGS. 7 and 8illustrate the back-trace operation and the resulting embedded path.

The A* search is not suitable for finding the lowest-cost path in agraph with non-zero dimensional states. This is because the A* searchcomputes a single cost value for the expansion to any state in thegraph, while the actual cost can vary across a non-zero dimensionalstate. Accordingly, there is a need for a path search process that canidentify the lowest-cost path in a graph with non-zero dimensionalstates.

SUMMARY OF THE INVENTION

Some embodiments of the invention provide a method for identifying apath between a set of source states and a set of target states in aspace with more than two dimensions. In some embodiments, the spaceincludes a set of states and a transition map that specifies a set ofstates that can be reached from each particular state. At least some ofthe states are non-zero dimensional states. For instance, in someembodiments, the space is a graph that includes points, lines, andsurfaces.

The method specifies at least one path that starts from one state. Itthen iteratively specifies new paths by expanding previously specifiedpaths to other states in the space until identifying a path thatconnects the source and target states. At least one of the expansions ofa previously specified path includes an expansion in more than twodimensions of the space.

BRIEF DESCRIPTION OF THE DRAWINGS

The novel features of the invention are set forth in the appendedclaims. However, for purpose of explanation, several embodiments of theinvention are set forth in the following figures.

FIGS. 1–8 provide an example of an A* path search that uses such an{circumflex over (F)} cost.

FIG. 9 illustrates a portion of a layer that has been triangulated intofour triangular faces, each of which has a space, three nodes, and threeedges, while FIG. 10 provides examples of topological routes that areformed by via nodes, Steiners, walls, joints, and nodes.

FIGS. 11 and 12 illustrate examples of line and surface PLF's.

FIG. 13 presents an example of filtering a first filtered PLF by asecond filter PLF, where both PLF's are defined across a line. FIG. 14illustrates the minimum PLF for the two PLF's of FIG. 13.

FIG. 15 illustrates a Q* path search process of some embodiments of theinvention.

FIGS. 16–19 provide examples for describing the process of FIG. 15.

FIG. 20 illustrates a process that propagates a PLF that is defined overa point to a line or a surface. FIGS. 21 and 22 illustrate examples ofpropagating a PLF from a point P to a line L and to a surface S.

FIG. 23 illustrates a process for propagating a PLF from a line toanother line or from a surface to a line, and FIGS. 24 and 25 providesexamples for describing this process.

FIGS. 26–31 illustrate how some embodiments identify the propagationvectors that emanate from the knot locations of a line PLF or surfacePLF.

FIG. 32 illustrates a process for propagating a G PLF from a line to asurface.

FIG. 33 illustrates an example for propagating a G PLF from a line to asurface.

FIG. 34 illustrates a process for propagating a PLF from a line to apoint or from a surface to a point. FIGS. 35 and 36 describe the processof FIG. 34.

FIG. 37 presents an example that illustrates an expansion from a startsurface, to a destination surface.

FIGS. 38A and 38B illustrate how to compute the flow across an edgeafter a potential expansion.

FIGS. 39 and 41 illustrate processes that generate PLF's that expresscosts of expansions to an edge or a hole, while FIGS. 40 and 42 presentexamples that describe these processes.

FIGS. 43–45 illustrate how to add two surface PLF's.

FIG. 46 illustrates a process that performs filtering and minimumoperations for an expansion to a line.

FIGS. 47–49 illustrate how to filter two surface PLF's.

FIGS. 50 and 51 illustrate how some embodiments compute an H function.

FIG. 52 illustrates a conceptual diagram of a computer system that isused in some embodiments.

DETAILED DESCRIPTION OF THE INVENTION

In the following description, numerous details are set forth for purposeof explanation. However, one of ordinary skill in the art will realizethat the invention may be practiced without the use of these specificdetails. In other instances, well-known structures and devices are shownin block diagram form in order not to obscure the description of theinvention with unnecessary detail.

Some embodiments of the invention provide a path search process that canidentify the lowest-cost path between source and target states in amulti-layer graph with non-zero dimensional states. Such a pathtraverses through intermediate states often but not always. In someembodiments described below, a zero-dimensional state is a point, aone-dimensional state is a line, a two-dimensional state is a surface, athree-dimensional state is a volume, etc.

The invention's path search can be used in many different applications.However, the embodiments described below use the path search in atopological router that defines topological routes for nets in a designlayout of an integrated circuit. Accordingly, in these embodiments, themulti-layer graph that the invention's path search explores is atessellated region of an IC layout. In this tessellated region, thesource, intermediate, and target states for a path search are zero-,one-, or two-dimensional topological particles. In other embodiments,the source, target, and intermediate states might be higher dimensionalstates (e.g., three-dimensional volumes, etc.).

The invention's path search operations can be used in different types ofpath searches, such as breadth-first searches, best-first searches, etc.In the embodiments described below, however, the invention's path searchis described as a best-first search. This best-first search is referredto below as the Q* search. The following sections provide (i) anoverview of the tessellated layout region that the Q* search explores,(ii) an overview of the Q* search, (iii) the overall flow of the Q*search, (iv) an example of a Q* search, (v) a description of the Q*search's costing of an expansion about a path, (vi) a description of themathematical operations used by the Q* search, (vii) a description of anĤ function computed by the Q* engine.

I. Overview of the Region for the Path Search

In the embodiments described below, the Q* search process is used in atopological router that defines topological routes that connect theroutable elements (e.g., pins) of nets in a region of a design layout.One such topological router is described in U.S. patent applicationentitled “Method and Apparatus for Routing Nets in an Integrated CircuitLayout”, having the Ser. No. 10/215,563 and filed on Aug. 9, 2002. Thisapplication is incorporated in the present application by reference.

The topological router that is described in the above-incorporatedapplication initially tessellates the IC-layout region, and then embedstopological routes in the tessellated region. At times, this routerfurther tessellates the region after embedding a route. The topologicalrouter can tessellate (i.e., decompose) the region in different types ofpolygons, such as rectangles, etc. In the embodiments described below,each layer of the IC-layout region is decomposed into triangular faces.The above-incorporated application describes in detail the triangulationand embedding operations. However, the triangulation and embeddingoperations are briefly described below, to describe the tessellatedregion that Q* search explores in some embodiments.

A. Triangulated Region after the Initial Triangulation

The initial triangulation results in numerous triangular faces. Eachresulting face has 3 edges, 3 nodes, a space representing thetopological regions inside the face, and possibly a set of holesrepresenting potential vias within the space. Nodes, edges, spaces, andholes are illustrated in FIG. 9, which presents a portion of anIC-layout layer after the initial triangulation. This figure illustratesfour triangular faces, each with three nodes, three edges between thethree nodes, and a space.

Some embodiments define nodes at the four corners of the layout regionon each layer, at the vertices of the obstacle and pin geometries in theregion, and at some or all virtual pins in the region (where virtualpins or “vpins” are points defined at the boundary of the region toaccount for the propagation of previously defined global routes into theregion). Edges are defined between certain pairs of nodes in order totriangulate the region. Some embodiments define each edge as an orderedlist of topological particles in the tessellated region. After theinitial triangulation, each edge's ordered list starts and ends with anode, and has a topological particle, called an “exit,” between the twonodes. For each edge, FIG. 9 illustrates two nodes and an exit betweenthe two nodes.

The topological router defines a hole between each overlapping pair ofspaces that have sufficiently large overlap to accommodate a via. Todetermine whether two spaces have sufficient overlap, the topologicalrouter maintains for each space a polygon that specifies the legal areawhere the center of vias can be in the space. The topological routerthen identifies the intersection of the polygons of each pair ofoverlapping spaces. For each overlapping space pair that has asufficiently large polygon intersection, the router then specifies ahole to represent a potential via that can be anywhere in the identifiedpolygonal intersection. A hole is part of both spaces that it connects.A space can contain several holes, one for each sufficiently overlappingspace in an adjacent layer. FIG. 9 illustrates holes in several spaces.The topological router performs its hole-identification process for aspace each time it creates or modifies a space. The above-incorporatedapplication further described how some embodiments identify holes.

B. Triangulated Region with Embedded Routes

After the initial triangulation, the topological router embedstopological routes in the decomposed topological region. The topologicalrouter defines a topological route as an ordered list of topologicalparticles in the decomposed topological region. Each embeddedtopological route is a graph that has several vertices and one or morewire segments that connect some or all the vertices. In the embodimentsdescribed below, the wire segments are referred to as walls and thevertices can be nodes, holes, “joints,” “via nodes,” and “Steiners.”

A “joint” is a topological particle that the topological router definesalong an edge (i.e., in the edge's ordered list) when this router embedsa path that crosses the edge. A joint divides a previously undividedexit of an edge into two exits. Via nodes are topological particles thatthe router creates when it embeds a topological path (all or some of atopological route) that utilizes a hole. Specifically, to embed a pathof a net that uses a hole, the router converts the hole into two vianodes, one in each space connected by the hole, and establishes anassociation between the two via nodes (e.g., has them refer to oneanother). A Steiner is a Steiner point (i.e., a vertex that connects tomore than two walls) that is generated at the juncture between apre-existing path of a net and a newly embedded path of the net. Whenthe router embeds a path that crosses an edge, it defines a joint at theintersection of the edge and the net's path. A joint divides apreviously undivided exit of an edge into two exits.

FIG. 10 illustrates two embedded topological paths 1000 and 1002. Bothof these paths traverse several triangular faces on layer 2. Path 1000also traverses two triangular faces on layer 3, as it is a multi-layerroute. Path 1000 is specified by the following ordered list ofparticles: node 1004, wall 1006, joint 1008, wall 1010, joint 1012, wall1014, via node 1016, via node 1018, wall 1020, joint 1022, wall 1024,and node 1026. Path 1002 is specified by the following ordered list ofparticles: node 1028, wall 1030, joint 1032, wall 1034, joint 1036, wall1038, Steiner 1040, wall 1042, and node 1044. FIG. 10 also illustrates awall 1046 of another portion of path 1002's route.

As shown in this figure, the embedded routes divide the faces that theycompletely cross into several spaces. For instance, these paths dividethe face formed by vertices 1004, 1048, and 1050 into three spaces.Also, this figure illustrates that each joint is at the intersection ofa route with an edge, and abuts two exits on its edge. The Steiner 1040of path 1002 connects walls 1038, 1042, and 1046. In addition,multi-layer route 1000 includes two via nodes 1016 and 1018. These vianodes enable this path to traverse layers 2 and 3. The topologicalrouter generated these two via nodes from the hole 905 of FIG. 9 when itembedded the path 1000. FIG. 10 also illustrates that a wall isspecified between each to non-wall, non-via particles on a path.

Each net has a representation of its connectivity and topology. Eachedge has an ordered list of the routes that cross it. In other words, aroute's topological particles that are on edges appear on their edge'sordered list (e.g., linked list) in the topological order that theirroute crosses their edge. For the example illustrated in FIG. 10, thetopological router specifies the edge between nodes 1048 and 1050 as anordered list of the following particles: node 1050, exit 1052, joint1008, exit 1054, joint 1036, exit 1056, and node 1048. In this edge'sordered list, joints 1008 and 1036 of paths 1000 and 1002 appear in theorder that their routes cross this edge.

Every particle that is within a face or is on an edge of the face isassociated with one or more spaces of the face. Some particles can bepart of multiple spaces. Each space contains its face's nodes and exitsjust after the initial triangulation. The topological router stores eachedge's capacity and length, and also stores the wire flow (route flow)across each edge.

In the embodiments described below, certain particles (such as joints,via nodes, walls, and Steiners) that define a route are moveable, whileother route-defining particles (such as pin-geometry nodes) have a fixedlocation. The topological router does not need to specify and store thelocation of moveable route-defining particles, as the routes aretopologically defined. However, in the embodiments described below, thetopological router specifies and stores locations for the route-definingparticles in order to be able to compute wirelength accurately.

II. Introduction to the Q* Path Search

The Q* path search can identify the lowest-cost path between source andtarget states in a multi-layer graph with non-zero dimensional states.In the embodiments described below, the source, intermediate, and targetstates in the graph are zero-, one-, or two-dimensional topologicalparticles in the tessellate region. In other words, these states arepoints (i.e., nodes), lines (i.e., edge- and wall-segments to which apath can expand), and surfaces (i.e., portions of holes polygons towhich a path can expand).

In some embodiments, the Q* search process has two phases: (1) a pathexploration phase, during which the process identifies a path betweenthe specified source and target particles, and (2) a path-embeddingphase, during which the process embeds the identified path. During thepath exploration phase, the Q* search starts at one or more paths fromthe source and/or target particles. It then iteratively identifies oneor more expansions about the lowest cost path, until it identifies apath that connects the source and target states. Each identifiedexpansion about a path is from a “current particle” (also called “startparticle”), reached by the path, to a “destination particle” thatneighbors the current particle.

The embodiments described below cost each expansion based on an F costthat can be expressed as:{circumflex over (F)}=G+Ĥ.  (1)The {circumflex over (F)}, G, and Ĥ are functions that are defined overthe entire destination particle of an expansion or one or more portionsof this destination particle. Specifically, the {circumflex over (F)}cost is a function that expresses the estimated cost of a path thattraverses from a source through the expansion's destination particle toa target. The G cost is a function that expresses the cost of the paththat has reached the expansion's destination particle. The Ĥ cost is afunction that expresses an estimated cost of a path from the expansion'sdestination particle to the target set. In the embodiments describedbelow, the Ĥ cost function expresses the lower-bound estimate of theshortest path from the expansion's destination particle to the targetset. Accordingly, in these embodiments, the {circumflex over (F)} costfunction expresses the estimated cost of a lowest-cost path from thesource through the expansion's destination particle to a target. Also,in these embodiments, the G function and hence the {circumflex over (F)}function account for several different types of path costs, such as adistance cost, route-piercing cost, via cost, wire-shoving costs, etc.Hence, the {circumflex over (F)} function of equation (1) allows theembodiments described below to identify the lowest-cost path between thesource and target sets. Other embodiments, however, might utilize adifferent {circumflex over (F)} function, as further described below.

In the embodiments described below, the G, Ĥ, and {circumflex over (F)}functions are convex piecewise linear functions (“PLF”), although inother embodiments they might be other types of functions. In theembodiments described below, each PLF's domain is either a point, aline, or a surface in the tessellated region. In the discussion below,PLF's that are defined over a point are called point PLF's, PLF's thatare defined across lines are called line PLF's, and PLF's that aredefined across surfaces are called surface PLF's.

A point PLF maps the point over which it is defined to a single value. Aline PLF is a PLF with a domain that is a line. Such a PLF can bespecified by a sequence of vertices, called knots, that represent theendpoints of its line segments. FIG. 11 illustrates an example of a linePLF 1110 that has four line segments. This PLF specifies a PLF-value foreach point P that is offset by an amount Q along an edge 1105 (i.e., foreach real number Q that defines a point P on the edge at Q·u, where u isa unit vector of L). Each knot of a line PLF can be specified in termsof an offset value Q and a PLF-value V. Also, the knots of a line PLFcan be sorted in an order based on their offset values. Accordingly, thefive knots K1–K5 that are at the endpoints of PLF 1110's four linesegments can represent this PLF of FIG. 11. These five knots can berepresented by the following ordered list of offset and PLF-value pairs:(Q1, V1), (Q2, V2), (Q3, V3), (Q4, V4), (Q5, V5).

A surface PLF is a set of one or more planar surfaces, called facets.FIG. 12 illustrates an example of a surface PLF. This PLF 1210 has fourfacets (F1–F4), each of which has a different slope. This PLF is definedacross a surface 1205. For each x-y value in the surface 1205, thesurface PLF 1210 provides a PLF-value (V).

Each surface PLF can be represented by a set of knots, a set of edgesbetween the knots, and a set of facets defined by the edges. Using thisapproach, the surface PLF 1210 of FIG. 12 can be represented by a listof knots K1–K10, a list of edges E1–E13, and a list of facets F1–F4.Some embodiments represent (1) a surface-PLF knot with an x,ycoordinate, a PLF-value at that coordinate, and a list of edges that areincident upon the knot, (2) a surface-PLF edge by a pair of referencesto two knots that the edge connects, and (3) a surface-PLF facet by alist of edges, a normal vector (e.g., x, y, z coordinate values), and az-intercept, where z is the axis in which the PLF-values are defined.For instance, in the example in FIG. 12, the list of edges for the knotK1 specifies edges E1 and E2, the list of knots for edge E4 specifiesknots K3 and K4, and the list of edges for facet F1 identifies edgesE1–E4. For facet F1, a normal vector N1 and a z-intercept are alsospecified.

In the embodiments described below, the Q* search maintains anotherfunction for each topological particle that can serve as a source,intermediate, or target state in the tessellated region. This functionis called a filter function. In the embodiments described below, thefilter function is a PLF. A particle's filter function expresses thelowest path cost that the search process has been able to identify fromthe source set to each point on the particle during a path search. Asfurther described below, the Q* search in the embodiments describedbelow uses the particle filter functions to determine whether apotential expansion to a destination particle is a viable one (i.e.,whether the expansion provides a cheaper path to any portion of thedestination particle than previously identified expansions to thedestination particle). This search makes this determination for anexpansion when it initially identifies the expansion and also later ifit expands about it.

Filtering is when a first PLF F1 (a filtered PLF) is compared with asecond PLF F2 (a filter PLF) to determine whether any portion of the F1needs to be discarded. The filter and filtered PLF's have overlappingdomains (i.e., domain(F1)∩domain(F2)≠null). In the embodiments describedbelow, filtering discards the portion of the filtered PLF F1 that islarger than the corresponding portion of the filter PLF (i.e., discardsevery portion of F1 where F1(V)>F2(V)). FIG. 13 presents an example offiltering a first filtered PLF 1305 by a second filter PLF 1310, whereboth PLF's are defined across a line 1350. Each PLF is represented byits sequence of knots, which is sorted by the domain-offset values.Knots K1–K5 specify the first PLF while knots K6–K10 specify the secondPLF 1310. As shown in this figure, this filtering discards portion 1315of PLF 1305 as the PLF-values of this portion are larger than thePLF-values of the corresponding portion 1320 of filter PLF 1310. Afterthe filtering operation, two portions 1325 and 1330 of the PLF 1305remain. Knots K1–K3 and K11 specify the first remaining portion 1325,while knots K12 and K5 specify the second remaining portion (where knotsK11 and K12 are at the intersection of the PLF's 1305 and 1310).

Filtering PLF's will be further described below. The filtering that isdescribed below not only filters the first PLF, but also defines thesecond filter PLF to be the minimum of the first and second PLF's. Aminimum of two PLF's F1 and F3 is another PLF F3 that specifies aPLF-value for each location V in the intersection of F1's domain andF2's domain that is equal to the smallest PLF-value specified by thefirst and second PLF's for that location (i.e., F3(V)≡min(F1(V),F2(V)).FIG. 14 illustrates the minimum PLF 1405 for the two PLF's 1305 and 1310of FIG. 13. The portion 1325 and 1330 of this minimum functioncorresponds to the remaining portion of the filtered PLF 1305, while theportion 1320 of this minimum function is from the original filter PLF1310. Knots K1–K3, K11, K8, K9, K12, and K5 specify the minimum function1405.

III. Overall Flow of the Q* Path Search Process

For some embodiments, FIG. 15 illustrates a Q* search process thatidentifies and embeds the lowest-cost path between specified source andtarget particles in the tessellated region. The process 1500 initiallysets (at 1502) a variable Current_Drop to null. The process 1500 usesdrops to represent path expansions. Specifically, this processrepresents an expansion from one topological particle (called a “startparticle” or “current particle”) to another topological particle (calleda “destination particle”) by a “drop” that associates with thedestination particle and refers back to the drop of the start particle.One of ordinary skill will realize that other embodiments might not usedrops or might implement drops differently.

At 1504, the process 1500 initializes filter and Ĥ PLF's for eachtopological particle in the tessellated region that can serve as astart, intermediate, or target particle for a path search. For each suchtopological particle, the process 1500 maintains (1) the filter PLF toexpress the lowest path cost that the process has been able to identifyfrom the source set to the particle during a path search, and (2) the ĤPLF to express the estimated distance between the particle and thetarget set. The process 1500 stores the H PLF for each particle so thatit only has to compute the Ĥ PLF once for each particle. In someembodiments, the process initializes the filter and H PLF's to“infinite”. Also, in the embodiments described below, nodes, exits, andholes are the topological particles that can serve as start particlesduring a path search, while nodes, holes, exits, and walls are thetopological particles that can serve as intermediate and targetparticles during the path search.

Next, for each source particle of the path search, the process 1500 (at1506) identifies and sets the particle's H PLF and sets the particle'sfilter PLF to zero. The generation of a particle's H PLF is describedbelow. At 1506, for each source particle, the process 1500 alsospecifies a drop, defines the source particle as the drop's particle,defines the drop's prior drop as null, sets the drop's G PLF to zero,and sets the drop's F PLF equal to the source particle's H PLF. Theprocess stores the specified drops in a storage structure. In theembodiments described below, the storage structure is a priority queue(e.g., a heap) that is sorted based on the minimum of the F PLF of eachdrop.

At 1508, the process then retrieves from the priority queue a drop withthe smallest minimum F value. Next, the process filters (at 1510) theretrieved drop's G PLF with the filter PLF of this drop's particle. Asfurther described below, the process performs this filtering operationto ensure that the retrieved drop is still valid. After the process 1500stored the retrieved drop in the priority queue, it might have createdand stored other drops that represent cheaper expansions to theretrieved drop's particle. These cheaper expansions would have modifiedthe filter PLF of the retrieved drop's particle.

FIG. 16 illustrates one such situation. This figure illustrates a G PLF1605 of a first expansion to a line 1610. This G PLF is also the filterfunction of the line 1610 after the first expansion. FIG. 16 illustratesa G PLF 1615 of a second expansion to the line 1610. The secondexpansion's G PLF 1615 is smaller than the first expansion's G PLF 1605over the entire line 1610 except for the portion 1625 of the line. Afterthe second expansion is identified, the filter function of the line 1610is set to the minimum of the second expansion's G PLF and the filterfunction's original PLF (which, as mentioned above, is identical to thefirst expansion's G PLF). This new filter function is the PLF 1620 inFIG. 16.

Accordingly, after retrieving a drop from the priority queue, theprocess 1500 filters (at 1510) the drop's G PLF with its particle'sfilter function to ensure that the drop still represents the best validexpansion to one or more portions of its particle. In the exampleillustrated in FIG. 16, the filtering of the G PLF 1605 of the firstexpansion to (i.e., the first drop on) the line 1610 with the line'sfilter PLF 1620 after the second expansion results in the PLF 1630. ThisPLF 1630 corresponds to the portion of the original PLF 1605 of thefirst expansion that is smaller than the second-expansion's PLF 1615.This PLF 1630 is defined over a much smaller domain (i.e., segment 1625)than the original PLF 1605 for the first expansion. Filtering onefunction against another will be further described below.

At 1512, the process determines whether any portion of theCurrent_Drop's G PLF remains after the filtering operation. If theprocess determines (at 1512) that at least one portion of theCurrent_Drop's G PLF remains after the filtering, the process determines(at 1514) whether the filtering operation at 1510 resulted in two ormore convex pieces of the retrieved drop's G PLF.

If the filtering did not result in two or more convex pieces, theprocess specifies (at 1516) the retrieved drop's F PLF again, as someportions of this drop's G PLF might have been discarded because of thefiltering operation, and such a modification would discard some portionsof this drop's F PLF. Next, the process determines (at 1518) whether theretrieved drop's F PLF minimum is greater than the lowest F PLF minimumof the drops that are currently stored in the priority queue. If so, theprocess stores (at 1522) the retrieved drop again in the priority queue,and then transitions back to 1508 to select another drop. Otherwise, theprocess specifies (at 1520) the retrieved drop as the Current_Drop andthen transitions to 1532, which will be further described below.

If the process determines (at 1514) that the filtering at 1510 resultedin two or more convex pieces of the retrieved drop's G PLF, the processspecifies (at 1524) a drop for each remaining piece and sets theirparameters as follows. The process defines each specified drop'sparticle as the retrieved drop's particle. It also sets each specifieddrop's previous drop identically to the retrieved drop's previous drop(which might be null). The process also sets each specified drop's G PLFequal to the portion of the retrieved drop's G PLF for which the dropwas specified. It also sets each specified drop's F PLF equal to the sumof (1) the specified drop's G PLF, and (2) the portion of the retrieveddrop's H PLF with the same domain as the specified drop's G PLF.

At 1526, the process then determines whether any of the drops created at1524 has the lowest F PLF minimum of all the drops stored in thepriority queue. If not, the process stores (at 1528) the drops specifiedat 1524 in the priority queue based on the minimum of the F PLF of eachdrop. From 1528, the process transitions back to 1508 to select anotherdrop. On the other hand, if the process determines (at 1526) that atleast one drop specified at 1524 has the lowest F PLF minimum of all thedrops stored in the priority queue, the process identifies (at 1530) asthe Current_Drop a drop that was specified at 1524 and that has thelowest F PLF minimum. At 1530, the process also stores the remainingspecified drops in the priority queue based on the minimum of the F PLFof each drop.

From 1530, the process transitions to 1532. The process also transitionsto 1532 from 1520. At 1532, the process determines whether theCurrent_Drop's particle is a target. If not, the process tries (at 1534)to expand the path search about the Current_Drop. Specifically, at 1534,the process tries to identify one or more potential expansions about theCurrent_Drop. Some embodiments identify potential expansions about theCurrent_Drop by (1) identifying a set of valid spaces to which a pathcan expand from the Current_Drop's particle, and (2) identifyingdestination particles in each identified space. A valid space is onethat contains the Current_Drop's particle but does not contain theparticle of the prior drop in a path search (i.e., does not contain theparticle of the Current_Drop's previous drop). FIG. 17 illustrates anexample of a valid expansion space for a planar expansion. This figureillustrates a path search 1705 that has a last drop 1710 and asecond-to-last drop 1715. The particles of both these drops are part ofspace 1725. The particle of drop 1710 is also part of space 1720.Consequently, space 1720 is a valid expansion space for drop 1710, butspace 1725 is not. There can be multiple viable expansion spaces as aretrieved drop's particle (such as a node) can be in multiple spaces.One of ordinary skill will realize that other embodiments might identifypotential expansions about the Current_Drop differently. For instance,some embodiments might define a valid space for an expansiondifferently. One such embodiment might not require that theCurrent_Drop's particle to be part of the valid space.

After the process identifies (at 1534) a set of valid spaces to whichthe path can expand from the Current_Drop's particle, it identifies (at1534) potential destination particles in each identified space. In someembodiments, a path can expand towards exits, walls and holes, as wellas the selected net's nodes, in a valid expansion space. One of ordinaryskill will realize that other embodiments might specify other potentialexpansion particles in a valid expansion space.

In the example illustrated in FIG. 17, the path search can expand fromdrop 1710 to exit 1745, hole 1750, and wall 1725. Wall 1725 belongs to aroute 1730 of another net. The path generation process 1500 might allowa path expansion to the wall of another net's previously defined pathbecause it might allow later defined routes to rip up earlier definedroutes. In such cases, expanding to the walls of another net's path isassessed a penalty, as further described below.

In some situations, the process 1500 cannot identify (at 1534) anypotential expansions to another particle. However, when the processidentifies one or more potential expansions at 1534, the processperforms the following four operations at 1534. First, it specifies theH PLF of each potential expansion's destination particle if it had notpreviously been set for the path search. The generation of the H PLF isdescribed below. Second, the process specifies a G PLF for eachpotential expansion. The generation of the G PLF for an expansion (i.e.,the costing of an expansion) is described below. Third, it filters the GPLF of each potential expansion with the filter PLF of the expansion'sdestination particle. This filtering operation also sets the destinationparticle's filter PLF equal to the minimum of the filtered G PLF and thedestination particle's previous filter PLF. Filtering two PLF's isdescribed below. Fourth, the process specifies a drop for each convexpiece (if any) of a G PLF of a potential expansion that remains afterthe filtering operation. For each specified drop, the process (1) setsthe drop's G PLF equal to the remaining piece of the filtered G PLF forwhich the drop was created, (2) associates the drop with its expansion'sdestination particle, (3) sets the drop's previous drop to theCurrent-Drop, and (4) sets the drop's F PLF to the sum of the drop's GPLF and the portion of its destination particle's H PLF that is definedover the domain of the drop's G PLF. The process stores each drop itcreates at 1534 in the priority queue.

From 1534, the process transitions to 1536. The process also transitionsto 1536 if it determines at 1512 that no portion of a retrieved drop's GPLF remains after the filtering at 1510. At 1536, the process determineswhether the priority queue that stores the drops is empty. If so, theprocess has failed to find a path between the specified source andtarget sets. Accordingly, it returns (at 1538) a notification specifyingits failure and then ends. On the other hand, when the processdetermines (at 1536) that the priority queue is not empty, the processtransitions back to 1508 to retrieve the next lowest-cost drop from thispriority queue and then to perform the subsequent operations for thisdrop.

The process has found a path between the source and target sets when itdetermines (at 1532) that the Current_Drop's particle is a target. Inthis situation, the process transitions from 1532 to 1540. At 1540, theprocess 1500 performs a back trace operation to embed topologically theidentified path between the source set and the target. Starting at theCurrent_Drop on the target, this operation “back traces” the sequence ofdrops that reached the target and generates an ordered list oftopological particles that define the topological path. Generation ofsuch an ordered list entails creation of wall particles between eachpair of non-wall, non-via particles in the topological path, and canalso entail the creation of joints, Steiners, and via nodes. In someembodiments, this operation also specifies coordinates for eachtopological particle that it creates. The back trace operation isfurther described in the above-incorporated application.

One of ordinary skill will realize that the Q* path-generation processmight be implemented differently in other embodiments. For instance,some embodiments might utilize non-convex PLF's. Also, in someembodiments, the Ĥ cost function might not specify a lower bound on theshortest path between a drop's particle and a target set. In addition,some embodiments might compute the {circumflex over (F)} functionslightly differently. For instance, some embodiments might express the{circumflex over (F)} function as:{circumflex over (F)}=G+2*Ĥ.Such a function would bias the search process to expand about the dropsthat are closer to the target set. Alternative embodiments might expressthe {circumflex over (F)} function as:{circumflex over (F)}=G+Ĥ+Ĵ,where Ĵ represents the estimated computational effort needed to completethe path from the current drop. The embodiments that use alternative{circumflex over (F)} function do not satisfy the admissibilityrequirement (i.e., they do not produce consistently the lowest-cost pathbetween source and target sets). On the other hand, the embodiments thatuse the {circumflex over (F)} function of equation (1) do satisfy thisrequirement.IV. Example of the Q* Search Process

FIG. 18 presents an example of the Q* path search. In this example, theQ* search process is used to construct the lowest-cost path between asingle source node 1805 and a set of target nodes 1810 in a triangulatedgraph 1800. To simplify this example, the only source and targetparticles are nodes, and the only intermediate particles are edges.Also, this example accounts only for the distance costs, and ignoresspacing constraints on edges due to obstacles.

FIG. 18 illustrates several sets of expansions identified during a pathexploration phase. The expansions are represented by drops 1 a, 1 b, 2,3 a, 3 b, 4 a, 4 b, 5–7, 8 a, 8 b, 9, 10, and 11. In this example, theQ* path search starts by identifying and storing drops 1 a and 1 b aboutthe source node 1805. This process then successively identifies thesequence of drops 2, 3 a and 3 b, and 4 a and 4 b. It identifies thissequence by successively (1) identifying each of the drops 1 a, 2, and 3a as the drops in the priority queue with the lowest F function minimum,and (2) identifying viable expansions about each of the edges of thesedrops. Once the process creates and stores expansion drops 4 a and 4 bas viable expansions from drop 3 a, the process identifies drop 1 b asthe drop with the lowest F function minimum in the priority queue.Accordingly, it then successively identifies the sequence of drops 5, 6,7, 8 a, 8 b, 9, 10, and 11 that reaches the target set 1810. The processidentifies this sequence by successively (1) identifying each of thedrops 5, 6, 7, 8 a, 9 as the drops in the priority queue with the lowestF function minimum, and (2) identifying viable expansions about the edgeof each of these drops.

In this example, two expansions are identified to each of the followingfour edges 1825, 1830, 1835, and 1840. The search process identifies thefirst expansions (i.e., identifies drops 3 a, 3 b, 4 a, and 4 b) tothese edges when it expands about drops 1 a, 2, 3 a, 3 b, and 4 a, whileit identifies the second expansions (i.e., identifies drops 6, 7, 8 a,and 8 b) to these edges when it expands about drops 5, 6, and 7. Thesecond set of drops (i.e., drops 6, 7, and 8 b) that are identified forthe edges 1825, 1830, and 1840 are only specified over portions 1845,1850, and 1855 of these edges. This is because the filtering operationsthat were performed to assess the viability of these second expansionsto edges 1825, 1830, and 1840 resulted in the filtering of their G PLF'soutside of the portions 1845, 1850, and 1855. However, the second drop 8a on edge 1835 is defined over the entire edge, as the filteringoperation that the process performs before identifying this drop doesnot filter any portion of this expansion's G PLF. This is because theentirety of the edge 1835 can be reached more cheaply through thesequence of drops 1 b, 5, 6, and 7 than through the sequence of drops 1a, 2, and 3 a. In the example illustrated in FIG. 18, the lowest-costpath is the one represented by drops 11, 9, 8 a, 7, 6, 5, 1 b, and 12,where drop 12 is the drop created for the source node 1805. Afteridentifying this sequence of drops, the Q* search process embeds a pathfor this sequence. An approximation of this embedded path is illustratedin FIG. 19. As shown in FIG. 19, joints are defined at the intersectionof the embedded path and the edges that this path intersects.

V. Computing G PLF

When the process 1500 identifies (at 1534) a potential expansion fromthe Current_Drop's particle to a destination particle, this processspecifies (at 1534) a G PLF for such a potential expansion. In someembodiments, the process 1500 computes this G PLF by (1) propagating theCurrent_Drop's G PLF to the destination particle, and (2) for certainexpansions, adding one or more penalty costs to the PLF resulting fromthe propagation.

A. Propagation

Propagating the Current_Drop's G PLF to an expansion's destinationparticle generates a PLF that expresses the distance and penalty costsof reaching the Current_Drop's particle plus the distance cost ofreaching the expansion's particle from the Current_Drop's particle. Inother words, the propagation accounts for the extra distance cost ofgoing from the Current_Drop's particle to the expansion's particle.

Some embodiments limit each propagation operation to only the portion ofthe expansion's destination particle to which a path can expand from theCurrent_Drop. The discussion below uses the phrase “destination state”to refer to the portion of an expansion's destination particle to whicha path can expand from the Current_Drop. This discussion also uses thephrases “the Current_Drop's domain” or “the start domain” to refer tothe domain of the Current_Drop's G PLF.

Nine different propagation operations are described below for ninepair-wise combinations of expansions between points, line, and surfaces.In these propagation operations, points represent nodes, lines representedge- and wall-segments to which a path can expand, and surfacesrepresent portions of holes to which a path can expand.

As mentioned above, a hole is specified between each pair of overlappingspaces when the intersection of the two polygons of the overlappingspaces is larger than a threshold size. Such a hole represents apotential via that can be anywhere in the polygonal intersection of thetwo polygons of the two overlapping spaces. In some embodiments, a pathfor a net can expand to each portion of a hole. These embodiments mightspecify the same width and spacing constraints for all nets being routedat a given time, or might construct each hole for the worst case ofconstraints (e.g., construct the polygons of the hole's overlappingspaces for the worst case of constraints). Alternatively, someembodiments construct each hole based on the best case of constraints(e.g., construct the polygons of the hole's overlapping spaces for thebest case of constraints). For an expansion of a path of a net to ahole, these embodiments then identify a sub-set of the hole's polygon(i.e., the polygonal surface represented by the hole) to which the net'spath can expand given the spacing constraints for the net. One mannerfor identifying the sub-set of hole polygons is described in theabove-incorporated application.

1. Expansion from Point to Point

To perform a propagation operation for an expansion that goes from anexpansion start point to an expansion destination point (e.g., goes fromone node to another node), the process 1500 identifies the distancebetween the two points and adds this distance to the G PLF of the startpoint (i.e., to the Current_Drop's G PLF). In some embodiments, theprocess measures the distance between two points in an IC layout usingonly a specified set of wiring directions that are specified by thelayout's wiring model. For instance, when two points are on anoctilinear layer that can have horizontal, vertical, and ±45° diagonalinterconnect lines, the distance between the two points is the shortestdistance that can be traversed by using horizontal, vertical, and ±45°diagonal interconnect lines. As disclosed in U.S. patent applicationentitled “Hierarchical Routing Method and Apparatus that UtilizeDiagonal Routes,” and having Ser. No. 10/013,813, the shortest distancebetween two points for such an octilinear wiring model:Distance=L+S*(sqrt(2)−1),  (2)where L and S respectively represent the long and short sides of arectangular bounding box that is aligned with the layout's x-y axes andthat encompasses the two points. This manner of computing the shortestdistance does not disfavor or penalize any preferred wiring directionover another preferred wiring direction.

Numerous other operations below require the computation of the distancebetween two points. The embodiments described below compute each suchdistance according to the above-described equation (2).

2. Expansion from Point to Line or Point to Surface

FIG. 20 illustrates a process 2000 that propagates a PLF that is definedover a point to a line or a surface. This process is described below byreference to FIG. 21, which illustrates an example of propagating a PLFfrom a point P to a line L, and FIG. 22, which illustrates an example ofpropagating a PLF from a point P to a surface S.

As show in FIG. 20, the process 2000 initially projects (at 2005)vectors in all available interconnect directions from the Current_Drop'spoint. These vectors are referred to below as propagation vectors. Theembodiments described below utilize a wiring model that specifieshorizontal, vertical, and ±45° diagonal interconnect lines on eachlayer. Accordingly, these embodiments project vectors in the 0°, 45°,90°, 135°, 180°, 225°, 270°, and 315° directions. FIGS. 21 and 22illustrate eight propagation vectors emanating from their points P in0°, 45°, 90°, 135°, 180°, 225°, 270°, and 315° directions. Someembodiments do not project vectors in all interconnect directions.Instead, they project only the propagation vectors that will intersectthe destination state. These propagation vectors are the vectors thatfall within a triangle defined by the start-state point and the leftmostand rightmost vertices of the destination state.

At 2010, the process then identifies the intersection of the projectedpropagation vectors and the destination line or surface. A propagationvector intersects with a line at a point that is the location of a knotof the destination's line PLF. FIG. 21 illustrates the intersection ofthe propagation vectors 2105, 2110, and 2115 at three points 2120, 2125,and 2130 along line L. A propagation vector intersects a surface either(1) at only a vertex of the surface, or (2) at two points on the surfaceboundary and an edge that runs through the surface. FIG. 22 illustratesthe intersection of two propagation vectors 2215 and 2220 with thesurface S along edges 2270 and 2275, which respectively terminate onboundary point pair 2225 and 2230 and point pair 2235 and 2240.

Next, for the expansion to the destination state, the process 2000 thenspecifies (at 2015) a G PLF with knots at the intersections identifiedat 2010. Specifically, for a destination line, the process 2000specifies (at 2015) a knot at each point of the destination line that apropagation vector intersects. FIG. 21 illustrates a G PLF 2145 withthree knots 2150, 2155, and 2160 for the three intersection points 2120,2125, and 2130. For a destination surface, the process 2000 specifies(at 2015) a knot at each surface vertex intersected by a propagationvector and a knot at each surface boundary point intersected by apropagation vector. FIG. 22 illustrates a G PLF 2260 with four knots2280, 2282, 2284, and 2286 that are defined for four boundaryintersection points 2225, 2230, 2235, and 2240.

At 2015, the process 2000 sets the PLF-value of each knot that itspecifies at 2015. The PLF-value of a knot specified at 2015 equals (1)the Current_Drop's PLF-value at the start point, plus (2) the distancebetween the knot's x,y coordinates and the start point P, where thisdistance is measured along the direction of the propagation vector thatwas used to identify the knot. For example, in FIG. 41, the PLF-value ofknot 4155 equals the PLF-value at point P plus the distance D betweenpoint P and intersection point 4125 along the direction of propagationvector 4110. In FIG. 22, the PLF-value of knot 2286 equals the PLF-valueat point P plus the distance D between point P and intersection point2240 along the direction of propagation vector 2220.

At 2020, the process specifies knots for the expansion's G PLF at thelocation of any unexamined vertex of the destination state. The verticesof a destination line are its endpoints. A line endpoint is anunexamined vertex if none of the projected propagation vectors intersectthe destination line at the endpoint. The PLF-value of an unexaminedendpoint of a destination line equals (1) the Current_Drop's PLF-valueat the start point, plus (2) the distance (according to the equation(2)) between the start point and the endpoint. For instance, in FIG. 41,the process specifies two knots 4165 and 4170 at the endpoints 4175 and4180 of the line L, and specifies the PLF-value of each of these knotsas the PLF-value at point P plus the distance (according to equation(2)) between point P and each knot location 4175 or 4180.

The vertices of a destination surface are the vertices of the edges thatdefine the surface's external boundary. Such a vertex is an unexaminedvertex if none of the propagation vectors intersect the destinationsurface at the vertex. In FIG. 22, the destination surface S has fivevertices 2265, all of which are “unexamined” as they are not at theintersection of the propagation vectors and the surface. Accordingly,five knots 2290 are specified at the unexamined vertices 2265. ThePLF-value for each such knot equals (1) the distance (according toequation (2)) between the start point and the surface's vertex thatcorresponds to the knot, plus (2) the Current_Drop's PLF-value at thestart point. For instance, the PLF-value of knot 2290 a equals thePLF-value at the start point P plus the distance (according to equation(2)) between point P and vertex 2265 a.

For a destination surface, the process 2000 uses (at 2025) the knotsspecified at 2015 and 2020 to specify the edges and facets of thesurface PLF that is defined over the destination surface. For eachfacet, the process defines a normal and a z-intercept. Standardplane-defining techniques can be used to derive the normal andz-intercept values of a facet from three points on the facet. For eachfacet, the process 2000 identified at least three knots at 2015 and2020.

3. Line to Line or Surface to Line

FIG. 23 illustrates a process 2300 for propagating a PLF from a line toanother line or from a surface to a line. This process is described byreference to FIG. 24, which illustrates the propagation of a line PLFfrom a line 2405 to another line 2410, and FIG. 25, which illustratesthe propagation of a surface PLF from a surface 2502 to a line 2504.

As shown in FIG. 23, the process 2300 initially identifies (at 2305) thepropagation vectors that emanate from the locations on theCurrent_Drop's domain that are locations of knots in the Current_Drop'sG PLF. The identification of these propagation vectors is furtherdescribed below by reference to FIGS. 26–31. In FIG. 24, knots arelocated at points 2415 and 2420 on line 2405. In FIG. 25, knots arelocated at vertices 2506–2518 of surface 2502.

Next, at 2310, the process 2300 projects the propagation vectorsidentified at 2305. FIG. 24 illustrates the projection of fivepropagation vectors from knot-location 2415 and five propagation vectorsfrom knot-location 2420. In FIG. 25, three propagation vectors areprojected from each of the vertices 2506, 2512, and 2514, twopropagation vectors are projected from each of the vertices 2508 and2518, and one propagation vector is projected from each of the vertices2510 and 2516. In some embodiments, the process 2300 does not project(at 2310) vectors in all interconnect directions. Instead, it projectsonly the propagation vectors that will intersect the destination state.In these embodiments, this process projects propagation vectors thatfall within a triangle defined by the knot-location and the leftmost andrightmost vertices of the destination line or surface.

The process 2300 identifies (at 2315) the intersection of thedestination line and the propagation vectors that were projected at2310. FIG. 24 illustrates the intersection of the propagation vectors2430 and 2435 and the line 2410 at points 2450 and 2455. FIG. 25illustrates the intersection of the propagation vectors 2522, 2524, and2526 and the line 2504 at points 2530, 2532, and 2534.

Each intersection identified at 2315 is a knot in the expansion's G PLF.Accordingly, for the expansion to the destination line, the processspecifies (at 2320) a line G PLF with a knot at each intersectionidentified at 2315. At 2320, the process computes the PLF-value of eachknot specified at 2320. The PLF-value of each destination-state knotequals (1) the PLF-value of the Current_Drop's knot that was used toidentify the destination-state knot, plus (2) the distance between thex,y coordinates of the Current_Drop and destination-state knots, wherethis distance is measured along the projected propagation vector thatidentified the destination-state knot.

For instance, in FIG. 24, the PLF-value of knot 2472, which is definedat the intersection 2450 of the propagation vector 2430 and the line2410, equals (1) the PLF-value of the Current_Drop's G PLF at 2420 onthe line 2405, plus (2) the distance D1 between points 2420 and 2450along the direction of the propagation vector 2430. Similarly, in FIG.25, the PLF-value of knot 2524, which is defined at the intersection2532 of the propagation vector 2524 and the line 2504, equals (1) thePLF-value of the Current_Drop's G PLF at 2512 on the surface 2502, plus(2) the distance D2 between points 2512 and 2532 along the direction ofthe propagation vector 2524.

At 2325, the process specifies knots for the expansion's G PLF at thelocation of unexamined vertices of the destination line, and thenterminates. As mentioned above, an unexamined vertex of a destinationline is an endpoint of the line that does not intersect any of theprojected propagation vectors. An unexamined destination-line vertex canbe in either (1) a “wedge” that is defined by two propagation vectorsthat emanate from the same location on the Current_Drop's domain, (2) a“channel” that is defined by two parallel propagation vectors thatemanate from two different locations on the Current_Drop's domain. ThePLF-value of a knot specified for an unexamined vertex that is within awedge defined by two propagation vectors emanating from the samestart-state location equals (1) the PLF-value of Current_Drop's G PLF atthe start-state location, plus (2) the distance (according to theequation (2)) between the start-state location and the unexaminedvertex. On the other hand, the PLF-value of a knot specified for anunexamined vertex that is within a channel defined equals (1) the lengthof a line segment that is parallel to the two channel-defining vectorsand that starts at the unexamined vertex and terminates at the startdomain, plus (2) the PLF-value of the Current_Drop's G PLF at the pointthat the line segment terminates on the start domain. The line segmentterminates on the start domain on a second line segment that is betweenthe two knot locations from which the two channel-defining vectorsemanate. When the start domain is a surface, the second line segment (1)is an edge on the boundary of the surface if the two knot locations areboundary vertices of the surface, and (2) is a line segment within thesurface if the two channel-defining knot locations are within thesurface.

For instance, in FIG. 24, endpoint 2445 of the line 2405 falls within achannel defined by propagation vectors 2425 and 2430. The distancebetween endpoint 2445 and line 2405 is the length D3 of a line segment2440 that is parallel to vectors 2425 and 2430. Accordingly, thePLF-value of the knot 2476 specified at endpoint 2445 equals the lengthD3 plus the PLF-value of the Current_Drop's G PLF at point 2480, whichis the location that line segment 2440 intersects line 2405. Theendpoint 2460, on the other hand, is within a wedge defined by twopropagation vectors 2435 and 2482 that emanate from the start-statelocation 2420 on the line 2405. Accordingly, the PLF-value for the knot2478 that is specified at 2460 equals (1) the PLF-value of theCurrent_Drop's G PLF at the start-state location, plus (2) the distancebetween points 2460 and 2420 according to the equation (2).

In FIG. 25, endpoint 2554 falls within a channel defined by propagationvectors 2520 and 2522. The distance between endpoint 2554 and surface2502 is the length D4 of a line segment 2560 that is parallel to vectors2520 and 2522. Accordingly, the PLF-value of the knot 2550 specified atpoint 2554 equals the length D4 plus the PLF-value of the Current_Drop'sG PLF at point 2562, which is the location that the line segment 2560intersects the surface boundary. The endpoint 2556, on the other hand,is within a wedge defined by two propagation vectors 2526 and 2564 thatemanate from the start-state location 2512. Accordingly, the PLF-valuefor the knot 2552 that is specified at 2556 equals (1) the PLF-value ofthe Current_Drop's G PLF at the start-state location 2512, plus (2) thedistance between points 2512 and 2556 according to the above-describedequation (2).

FIGS. 26–31 illustrate how some embodiments identify the propagationvectors that emanate from the knot locations of a line PLF or surfacePLF. These embodiments identify the propagation vectors based on thefollowing observations. At most eight propagation-vector wedges can bedefined about a Current_Drop's domain when the octilinear wiring modelthat provides horizontal, vertical, and ±45° diagonal interconnect lineson each layer is used. Knots can be the only sources for wedges.

As mentioned above, each wedge is defined by two propagation vectorsthat emanate from the same knot location. Two wedges are abutting wedgeswhen they share a propagation vector, while two wedges are adjacentwedges when they have parallel propagation vectors. For instance, FIG.26 illustrates eight wedges 2650–2664 that are defined about five knotlocations 2625–2645 from a surface PLF's domain 2600. In this figure,there are three abutting wedge pairs, and five adjacent wedge pairs. Forinstance, wedges 2658 and 2656 are abutting wedges as they share vector2666, while wedges 2654 and 2656 are adjacent wedges as their vectors2668 and 2670 are parallel.

The parallel propagation vectors of two adjacent wedges define afreeway. For instance, a freeway 2672 exits between adjacent wedge pairs2662 and 2664 in FIG. 26. This freeway either (1) defines a channel whenno other propagation vectors emanates from a knot location that fallswithin the freeway, or (2) contains two or more channels when otherpropagation vectors that are parallel to the freeway emanate from knotlocation(s) that fall(s) within the freeway. At most there are eightadjacent wedge pairs about a Current_Drop's domain. Consequently, thereare at most eight freeways between the adjacent wedge pairs.

Some embodiments define the direction of the propagation vectors thatemanate from an a Current_Drop's domain by performing the followingthree operations. First, they identify the knot location for each wedge.Second, these embodiments identify one or more freeways between adjacentwedge pairs. Third, for each identified freeway, these embodimentsdetermine whether to treat the freeway as one channel, or to defineadditional channels within the freeway by defining one or morepropagation vectors that are parallel to the freeway-defining vectorsand that emanate from knot location(s) that fall within the freeway. Thefirst operation (i.e., the identification of the knot location for eachwedge) is described below by reference to FIGS. 27 and 28. The secondand third operations (i.e., the identification of freeways andadditional channels within the freeways) are then described by referenceto FIGS. 29–31.

FIG. 28 illustrates a process 2800 that identifies the knot location foreach propagation-vector wedge that emanates from the Current_Drop'sdomain, which can be a line or a surface. This process will be describedby reference to an example illustrated in FIG. 27. This exampleillustrates how one of the wedges of the PLF-surface domain 2600 of FIG.26 is identified.

The process 2800 initially selects (at 2805) a wedge. There are eightwedges when the octilinear wiring model that specifies horizontal,vertical, and ±45° diagonal interconnect lines on each layer is used.The process then selects (at 2810) a knot location as the firstcandidate location for the source of the selected wedge, and recordsthis knot location as the current best (Current_Best) location for theselected wedge. FIG. 27 illustrates a selected wedge 2660 and a firstcandidate knot location 2610. Next, at 2815, the process selects asecond candidate knot location for the source of the selected wedge andrecords it as Next_Candidate location. In FIG. 27, the second candidatelocation is location 2625.

At 2820, the process then determines whether the Current_Best knotlocation is a better candidate location than the Next_Candidatelocation. This determination ensures that all locations in the wedge arebest reached from the source of that wedge, when both distance andPLF-value is considered. To make this determination, the processperforms the following four operations. First, the process uses theselected wedge's “vector,” which in some embodiments is defined to be aunit vector that bisects the wedge (i.e., it is midway between the twovectors that define the wedge). FIG. 27 illustrates the vector 2705 ofthe selected wedge 2660.

Second, the process computes the dot product of the selected wedge'svector and a vector that starts from Current_Best knot location andterminates on Next_Candidate knot location. This computation quantifieswhether the Next_Candidate knot location is closer to an arbitrary pointin the wedge than the Current_Best knot location. FIG. 27 illustrates avector 2710 from the Current_Best location 2610 to the Next_Candidatelocation 2625.

Third, after computing the dot product, the process computes aCost_Delta, which is the difference in the PLF-values of Next_Candidatelocation and the Current_Best location according to the Current_Drop's GPLF (Cost_Delta=G PLF(Next_Candidate)−G PLF(Current_Best)). For theexample in FIG. 27, the Cost_Delta is the difference in the PLF-valuesat locations 2625 and 2610.

Fourth, the process determines whether the computed dot product isgreater than the Cost_Delta. If not, the Current_Best location is betterthan the Next_Candidate location, and the process transitions from 2820to 2830, which is further described below. If so, the Next_Candidatelocation is a better location for the wedge than Current_Best location,and the process transitions from 2820 to 2825. At 2825, the process setsthe Current_Best location equal to the Next_Candidate location, and thentransitions to 2830. At 2830, the process determines whether it hasexamined all knot locations for the wedge selected at 2805. If not, theprocess selects (at 2835) another knot location, sets this knot locationas the Next_Candidate, and transitions to 2820 to compare this newlyselected Next_Candidate with the Current_Best.

When the process determines at 2830 that it has examined all thelocations of knots in the Current_Drop's G PLF, the process defines (at2840) Current_Best as the knot location for the selected wedge. Theprocess then determines (at 2845) whether it has examined all thewedges. If not, it returns to 2805 to select another wedge and to repeatits operations for this wedge. Otherwise, it ends.

After identifying the locations of the wedges, the Q* engine has tospecify the channels between the wedges. When two wedges abut (i.e.,when they share a propagation vector), no channel can exist between thewedges. However, when two wedges are adjacent wedges (i.e., when theyhave parallel propagation vectors) one or more channels can be definedin the freeway defined by the parallel vectors of the adjacent wedgepairs.

If the Current_Drop's domain is a line, the Q* engine examines eachadjacent wedge pair that is defined along the line. If no knot locationexists on the line segment between the adjacent wedge pair, then the Q*engine defines the freeway between the adjacent wedge pair as a channel.If one or more knot locations exist on the line segment between anadjacent wedge pair, the Q* engine examines each knot location todetermine whether they are sources of propagation vectors. The engineinitially sorts in descending order the PLF-values of the knot locationsbetween the adjacent wedge pair. If there is only one knot locationbetween the adjacent wedge pair, the engine simply adds this PLF-valueto its sorted list. The engine then selects the largest PLF-value on thesorted list. The Q* engine then identifies the channel that contains theknot location corresponding to the selected PLF-value. This channel isthe freeway formed by the adjacent wedge pair when the knot location isthe first location between the wedge pair that is examined. When theknot location is not the first location between the wedge pair, thechannel might be formed by one or two propagation vectors that theengine specified for knot locations (between the wedge pair) that itpreviously examined.

The engine next determines whether the selected PLF-value for the knotlocation is less than the PLF-value that can be obtained by linearlyinterpolating between the values at the knot locations of the vectorsthat define the channel that contains the selected PLF-value's knotlocation (i.e., determines if the selected value is below a line thatconnects the PLF-values at the knot locations from which theidentified-channel's vectors emanate). If so, the engine specifies achannel-defining vector at the knot location associated with theselected PLF-value. This specified vector is parallel to the parallelvectors of the adjacent wedge pair.

FIG. 29 illustrates an example of a knot location 2930 that is betweentwo adjacent wedge pairs (where one pair is formed by wedges 2910 and2915 and one pair is formed by wedges 2920 and 2925). This knot locationis examined for each of these adjacent wedge pairs. When the engineexamines location 2930 for adjacent wedges 2910 and 2915, it determineswhether the PLF-value of this location is smaller than the PLF-valuethat can be obtained for this location by linearly interpolating betweenthe PLF-values at knot locations 2945 and 2950 that serves as theemanating location of the wedges 2910 and 2915. In FIG. 29, it isassumed that the PLF-value of location 2930 is less than the PLF-valuethat can be obtained through the linear interpolation. Accordingly, twopropagation vectors 2935 and 2940 are defined for the knot location2930. The propagation vector 2935 is parallel to the parallel vectors ofadjacent wedges 2910 and 2915, while the propagation vector 2940 isparallel to the parallel vectors of adjacent wedges 2920 and 2925.

FIG. 30 illustrates a process 3000 for defining channels betweenadjacent wedge pairs when the Current_Drop's domain is a surface. Thisprocess will be described by reference to FIG. 31A, which illustratesthe domain 3100 of a surface PLF. The process 3000 initially identifies(at 3005) all adjacent wedge pairs. It then selects (at 3010) one of theadjacent wedge pairs. In FIG. 31A, the selected pair of adjacent wedgesare wedges 3105 and 3110.

Next, at 3015, the process projects each location of a knot (in theCurrent_Drop's G PLF), which falls within the freeway that the selectedwedge pair defines, towards a line that connects the knot locations fromwhich the wedge pair emanates. The projection is in a direction that isparallel to the adjacent-wedge-pair vectors that define the freeway. InFIG. 31A, knot locations 3120, 3125, 3130, 3135, 3140, and 3142 fallwithin the freeway 3115 between the adjacent wedges 3105 and 3110. Also,the two wedges emanate from knot locations 3145 and 3150, and an edge3170 exists between these two locations. Knot location 3120 lies on theedge 3170, and hence does not need to be projected towards this linesegment. On the other hand, knot locations 3125, 3130, 3135, 3140, and3142 need to be projected towards this line segment in the direction ofthe vectors 3155 and 3160 that define the freeway 3115. FIG. 31Aillustrates the projection of knot location 3125 towards this edge.

At 3015, the process then computes a cost at the intersection of eachprojection with the line. The cost at each intersection point equals thesum of the PLF-value of the G PLF at the intersection point plus thedistance between the projected knot location and the intersection of theknot-location's projection and the line. In FIG. 31A, the intersectionof the projection of knot location 3125 is point 3165 on the edge 3150.Its computed cost equals the distance D between knot location 3125 andpoint 3165 and the value of the G PLF at the point 3165. The distance Dbetween a projected knot location and the intersection of its projectionand the line might be a positive or a negative value depending onposition of the knot location with respect to the line that connects theknot locations from which the adjacent wedge pair emanates. Forinstance, FIG. 31B illustrates an edge 3175 within the domain 3180 of asurface PLF. This edge connects to adjacent wedge pair 3182 and 3184.FIG. 31B also illustrates two knot locations 3186 and 3188 that fallwithin the freeway 3190 (defined by the wedge pair 3182 and 3184) onopposing sides of the edge 3175. The distance D1 between knot location3186 and this knot's projection point 3192 on the edge 3175 is positive,but the distance D2 between the knot location 3188 and this knot'sprojection point 3194 on the edge 3175 is negative. The distance D2 isnegative because the knot location 3188 is on the side of the edge 3175that is in the direction of the freeway-defining wedge vectors.

At 3020, the process sorts the computed PLF-values in descending order.It then selects (at 3025) the largest PLF-value on the sorted list. Theprocess then identifies (at 3030) the channel that contains theintersection point corresponding to the PLF-value selected at 3025. Thischannel is the freeway that the adjacent wedge pair selected at 3010defines at the first iteration through 3030 for the selected wedge pair.In the subsequent iterations through 3030, the propagation vectors thatthe process 3000 might identify for the knot locations between theadjacent wedge pair might define this channel partially or completely.

After 3030, the process determines whether the selected PLF-value forthe intersection point is less than the PLF-value that can be obtainedby linearly interpolating between the values at the two knot locationsof the vectors that define the channel that contains the intersectionpoint (i.e., determines if the selected value is below a line thatconnects the PLF-values at the knot locations of theidentified-channel's vectors). For instance, in FIG. 31A, the process3100 has the PLF-value at the knot locations 3145 and 3150. From thesetwo PLF-values the process can linearly interpolate the value of anypoint on the line 3150 between these two locations. Accordingly, if thefirst selected PLF-value is the PLF-value at the intersection point 3165for the knot location 3125, the process determines (at 3130) whether thecost computed for this knot location's projection onto point 3165 isless than the value that can be obtained for this point by linearlyinterpolating between the values specified at knot locations 3145 and3150.

If the process determines (at 3135) that the selected PLF value is notless than the value that can be obtained through the linearinterpolation, the process transitions to 3045, which is furtherdescribed below. Otherwise, the process specifies (at 3040) achannel-defining vector at the knot location associated with theselected PLF-value. This specified vector is parallel to the parallelvectors of the adjacent wedge pair. From 3040, the process transitionsto 3045.

At 3045, the process determines whether it has examined all the knotlocations that fall within the freeway defined by the adjacent wedgepair. If not, the process selects (at 3050) the next largest PLF-valuein the sorted list that it created at 3020. The process then performs3030 and 3035 for the intersection point that corresponds to thePLF-value selected at 3050. As mentioned above, the first iterationthrough 3030 identifies the freeway as the channel that contains theintersection point corresponding to the PLF-value selected at 3025.However, a subsequent iteration through 3030 from 3050 might identify adifferent channel that contains the intersection point corresponding tothe PLF-value selected at 3050. This is because, if the process iteratesthrough 3040 one or more times, it defines more propagation vectors thatbreak the freeway into smaller and smaller channels. When the process3000 identifies (at 3030) a smaller channel (i.e., a channel that coversonly a portion of the freeway) that contains the intersection pointcorresponding to the selected PLF-value, it derives (at 3035) theinterpolated value based on PLF-value of the knot locations from whichthe vectors that define the channel emanate.

When the process determines at 3045 that it has examined all the knotlocations that fall within the selected wedge pair's freeway, theprocess determines (at 3055) whether it has examined all the adjacentwedge pairs. If not, the process transitions back to 3010 to select anunexamined wedge pair. Otherwise, the process terminates.

4. Line to Surface

FIG. 32 illustrates a process 3200 for propagating a G PLF from a lineto a surface. The process 3200 is quite similar to the process 2300 ofFIG. 23. Accordingly, similar reference numbers are used for similaroperations of the two processes. The process 3200 has only a few minordifferences with the process 2300. Propagating a PLF to a surfaceresults in a surface PLF. Hence, after 2325, the process 3200 uses theattributes of the knots that it specifies to define edges and facets ofa surface PLF, and to define the normal and z-intercept values of thefacets.

Also, in process 3200, a propagation vector intersects the destinationsurface either at only a vertex or along an edge that runs through thesurface and connects two points on the boundary of the surface. For thecase where the propagation vector intersects the surface only at avertex, the process 3200 would specify (at 2320) a knot at the vertex'sx,y coordinates. For the case where the propagation vector runs throughthe surface, the process 3200 would specify (at 2320) two knots at thesurface boundary points where the propagation vector intersects thesurface boundary.

FIG. 33 presents an example that illustrates the propagation of a G PLFfrom a line 3305 to a surface 3310. For such a propagation, the process3200 would initially identify (at 2305) the propagation vectors thatemanate from the locations on the Current_Drop's line that are locationsof knots in the Current_Drop's G PLF. These propagation vectors areidentified based on the process described above by reference to FIGS.26–31. In FIG. 33, the knots are located at points 3315 and 3320 on line3305. From each of these points, five propagation vectors are projected.From these points, some embodiments would project only the propagationvectors that would intersect the destination surface, as mentionedabove.

Next, at 2310 and 2315, the process 3200 projects the propagationvectors identified at 2305, and identifies their intersections with thedestination surface. In FIG. 33, the propagation vectors 3325 and 3330that emanate from points 3315 and 3320 intersect the destination surfacealong edges 3360 and 3362, which respectively terminate on boundarypoint pair 3335 and 3340 and point pair 3345 and 3350.

The process 3200 then starts to specify (at 2320) a G PLF that isdefined over the destination surface. Specifically, at 2320, the processspecifies a knot at each intersection of the propagation vectors and theboundary of the destination surface. In FIG. 33, the process starts tospecify a surface PLF 3398 by specifying four knots 3352, 3354, 3356,and 3358 at the identified intersections 3335, 3340, 3345, and 3350. ThePLF-value of each specified destination knot equals (1) the PLF-value ofthe start knot that was used to identify the destination knot, plus (2)the distance between the x,y coordinates of the start and destinationknots, where this distance is measured along the projected propagationvector that identified the destination knot. For instance, the PLF-valueof knot 3354 that is specified for the location 3340 equals the distanceD1 between 3340 and 3315, plus the PLF-value of the Current_Drop's G PLFat 3315.

At 2325, the process 3200 then specifies knots for the destination's PLFat the location of the unexamined vertices of the destination. In FIG.33, the unexamined vertices of the destination surface are vertices3364, 3366, 3368, and 3370. For these vertices, the process 3200specifies knots 3372, 3374, 3376, and 3378. The PLF-value of each ofthese knots is computed based on whether the knot's correspondingdestination-surface vertex falls within a propagation-vector wedge orchannel.

For instance, the unexamined vertex 3370 falls within a wedge defined bytwo propagation vectors 3325 and 3322 that project from 3315.Accordingly, the PLF-value of knot 3378 at vertex 3370 equals (1) thePLF-value of the Current_Drop's G PLF at point 3315, plus (2) thedistance (according to equation (2)) between points 3315 and 3370. Onthe other hand, the unexamined vertex 3368 falls within a channeldefined by two parallel propagation vectors 3325 and 3330 that projectfrom two different points 3315 and 3320. The distance between the vertex3368 and the line 3305 is the length D2 of a line segment 3380 that isparallel to vectors 3325 and 3330. This line segment intersects line3305 at 3382. Hence, the PLF-value of knot 3376 at vertex 3368 equals(1) the length D2, plus (2) the Current_Drop's G PLF-value at 3382.

After 2325, the process 3200 uses (at 3205) the knots specified at 2320and 2325 to specify the edges and facets of the surface PLF that isdefined over the destination surface. For each facet, the processdefines a normal and a z-intercept. FIG. 33 illustrates three facets andten edges that define these three facets.

5. Surface to Point or Line to Point

FIG. 34 illustrates a process 3400 for propagating a PLF from a line toa point or from a surface to a point. This process is described byreference to FIG. 35, which illustrates the propagation of a line PLFfrom a line 3505 to a point 3520, and FIG. 36, which illustrates thepropagation of a surface PLF from a surface 3602 to a point 3620.

As shown in FIG. 34, the process 3400 initially identifies (at 3405) thepropagation vectors that emanate from the locations on theCurrent_Drop's domain that are locations of knots in the Current_Drop'sG PLF. The identification of these propagation vectors was describedabove by reference to FIGS. 26–31. In FIG. 35, knots are located atpoints 3510 and 3515 on line 3505. In FIG. 36, knots are located atvertices 3604–3616 of surface 3602.

Next, at 3410, the process 3400 projects the propagation vectorsidentified at 3405. FIG. 35 illustrates the projection of sixpropagation vectors from knot-location 3510 and four propagation vectorsfrom knot-location 3515. In FIG. 36, three propagation vectors areprojected from each of the vertices 3606 and 3614, two propagationvectors are projected from each of the vertices 3604, 3608, 3610, and3612, and one propagation vector is projected from vertex 3616.

The process 3400 then identifies (at 3415) the propagation-vector wedgeor channel that contains the destination point. As mentioned above, apropagation-vector “wedge” is defined by two propagation vectors thatemanate from the same location on the start domain, while a “channel” isdefined by two parallel propagation vectors that emanate from twodifferent locations on the start domain. After identifying the wedge orchannel in which the destination point falls, the process computes (at3415) the PLF-value at the destination point. The PLF-value at thedestination point that is within a wedge equals (1) the PLF-value ofCurrent_Drop's G PLF at the start domain vertex from which the wedge'spropagation vectors emanate, plus (2) the distance (according toequation (2)) between this vertex and the destination point. On theother hand, the PLF-value at a destination point that is within achannel equals (1) the length of a line segment that is parallel to thetwo channel-defining vectors and that starts at the destination pointand terminates at the start domain, plus (2) the PLF-value of theCurrent_Drop's G PLF at the point that the line segment terminates onthe start domain. The line segment terminates on the start domain on asecond line segment that is between the two knot locations from whichthe two channel-defining vectors emanate. When the start domain is asurface, the second line segment (1) is an edge on the boundary of thesurface if the two knot locations are boundary vertices of the surface,and (2) is a line segment within the surface if the two channel-definingknot locations are within the surface.

For instance, in FIG. 35, the destination point 3520 falls within achannel defined by two propagation vectors 3530 and 3535. The distancebetween the destination point 3520 and line 3505 is the length D of aline segment 3550 that is parallel to vectors 3530 and 3535.Accordingly, the PLF-value at destination point 3520 equals the length Dplus the PLF-value of the Current_Drop's G PLF at point 3545, which isthe location that line segment 3550 intersects line 3505. If thedestination point was point 3525 that is within the wedge defined bypropagation vectors 3530 and 3540, the PLF-value at point 3525 would be(1) the PLF-value of the Current_Drop's G PLF at point 3515, plus (2)the distance (according to equation (2)) between points 3515 and 3525.

In FIG. 36, the destination point 3620 falls within a channel defined bytwo propagation vectors 3624 and 3626. The distance between thedestination point 3620 and surface 3602 is the length D of a linesegment 3630 that is parallel to vectors 3624 and 3626. Accordingly, thePLF-value at destination point 3620 equals the length D plus thePLF-value of the Current_Drop's G PLF at point 3662, which is thelocation that line segment 3630 intersects surface 3602. If thedestination point was point 3618 that is within the wedge defined bypropagation vectors 3622 and 3624, the PLF-value at point 3618 would be(1) the PLF-value of the Current_Drop's G PLF at point 3610, plus (2)the distance (according to equation (2)) between points 3610 and 3618.

After 3415, the process 3400 terminates.

6. Expansion from Surface to Surface

A G PLF is propagated from a surface to another surface when, forexample, the expansion is from one hole to another hole. In theembodiments described below, such an expansion would define atopologically stacked via, which is a topologically defined via thatwould start and end on two non-adjacent layers. A topologically definedstacked via does not always result in a geometrically stacked via.

A viable expansion from one hole to another would require that the startand destination surfaces (i.e., the portions of hole polygons to which apath can expand) have sufficient overlap. Some embodiments define the GPLF of such an expansion only over the region of the destinationpolygonal surface that overlaps the start polygonal surface (i.e., onlyat the intersection of the domains of the start and destinationsurfaces). In this region, the expansion's G PLF would be identical tothe Current_Drop's G PLF for the corresponding region of the startpolygonal surface. Of course, the expansion's G PLF might have new knotsand might have modified edge and facet descriptions to account for theboundary of the overlapping regions.

FIG. 37 presents an example that illustrates an expansion from a startsurface to a destination surface. Two spaces in layers 2 and 3 mightcontain the start surface, while two spaces in layers 3 and 4 mightcontain the destination surface. FIG. 37 illustrates a start polygonalsurface 3705 and a destination polygonal surface 3710. This figure alsoillustrates a region 3715 that the polygons 3705 and 3710 overlap. Italso illustrates the G PLF 3720 that is defined over the start surface3705. This figure illustrates a projected view of the G PLF 3720 ontothe x,y plane, in order to simplify the visual presentation of this PLF.

In FIG. 37, the overlap region 3715 is sufficiently large. Hence, theexpansion is viable, and a G PLF needs to be computed for the expansion.FIG. 37 illustrates that the expansion's G PLF 3770 is defined only overthe overlapping portion 3715. This PLF 3770 is identical to the portionof the Current_Drop's G PLF 3720 that is defined across the overlappingregion 3715. However, the PLF 3770 has new knots and modified edge andfacet descriptions in order to account for the boundary of the overlapregion. In particular, the expansion's G PLF 3770 will include twofacets 3725 and 3730. Facets 3725 and 3730 have the same normal as thetwo facets 3740 and 3745 of the Current_Drop G PLF. However, thesefacets have slightly different knots and edges. Specifically, facet 3725includes knots 3750, 3754, 3756, 3764, and 3766, while facet 3730includes knots 3756, 3758, 3760, 3762, and 3764. All these knots specifythe boundaries of facets 3725 and 3730, which are smaller than thefacets 3740 and 3745 of the Current_Drop's PLF. Also, the edge 3772between facets 3725 and 3730 is shorter than the edge between facets3740 and 3745.

B. Penalty Cost

Propagating the Current_Drop's G PLF to a potential destination particlespecifies an initial G PLF for a potential expansion. If the potentialexpansion needs to be penalized, the process 1500 adds to the initial GPLF one or more penalty costs associated with the potential expansion.Some embodiments penalize expansions to holes, expansions toovercongested edges and expansions to walls of other nets. Someembodiments also penalize expansions that shove routes of other netsalong edges or within faces.

1. Penalty for Expansion to Overcongested Edges

Process 1500 allows a path to expand to the exits of overcongestededges. The route flow across an edge equals the width of the net routescrossing the edge plus the spacing between the crossing net routes andbetween the net routes and the edge nodes. FIGS. 38A and 38B illustratehow to compute the flow across an edge after a potential expansion.Specifically, FIG. 38A illustrates the center-lines of topologicalroutes for two nets that were previously inserted across an edge 3800.The two routes have widths W1 and W2 and spacing S1 and S3,respectively, towards their adjacent edge nodes. Also, the spacingbetween the two routes is defined as S2. FIG. 38B illustrates how theedge 3800 would look like after a third route is inserted across it. Theflow of this edge equals the sum of the following: (1) the minimumspacing S1 between net 1 and its adjacent node, (2) the width W1 of net1, (3) the minimum spacing S4 between nets 1 and 3, (4) the width W3 ofnet 3, (5) the minimum spacing S5 between nets 3 and 2, (6) the width W2of net 2, and (7) the minimum spacing S3 between net 3 and its adjacentnode.

When the route flow is equal to or less than the capacity of the edge,the expansion is allowed and legal. On the other hand, when this flow islarger than the edge capacity, the expansion is still allowed, but it isrecorded as an illegal expansion. Some embodiments compute the capacityof each edge according to the processes described above by reference toFIGS. 19 and 20.

Some embodiments pretabulate in a storage structure the widths for eachnet on each layer and the spacing requirements between each net andevery other net, pin, or obstacle on each layer. Accordingly, in theseembodiments, the path-generation process 1500 retrieves these valuesfrom the storage structure when it needs them (e.g., when it iscomputing edge capacities).

An expansion to an edge that is overcongested before or after theexpansion is assessed an extra penalty cost. This penalty is called acongestion penalty. In some embodiments, this congestion penalty is apre-defined penalty constant. In other embodiments, this penalty is aconstant that is derived based on the congestion of the edge. Thisconstant is linearly or non-linearly (e.g., quadratically) proportionalto the amount of over-congestion on the edge.

2. Expansion to Holes

Expansions to holes are also assessed a penalty cost. This penalty canbe different for holes between different layer pairs if the costintroduced by a via between different layer pairs is different. A viaintroduces a variety of costs in a design, and the penalty can take intoaccount all these costs. For instance, from a resistance or delay pointof view, a via might cost fifty times more than the resistance or delayof a unit length of a wire. Accordingly, the penalty might be 50 (i.e.,the via can be counted as a wire that is 50 units long). Alternatively,each via might reduce the manufacturing yield by some amount. Thisreduction can also be accounted for by equating it to a wirelength andadding it to the cost function.

3. Expansion to Walls of Other Nets

As mentioned above by reference to FIG. 17, the process 1500 allows thepath for one net to expand to the wall of another net (i.e., to specifya drop for an expansion to the wall of another net). The process,however, adds a rip penalty to the cost (i.e., G PLF) of an expansion toa wall of another net. This is because if such a drop is ever retrievedfrom the storage structure, the expansions that are defined about thisdrop will require piercing of the other net's route. In someembodiments, the rip penalty is larger than two via penalties. Such apenalty ensures that the path search only rips another net's route whenit has reasonably exhausted all other viable expansions on the samelayer or in the layers above or below.

4. Expansion that Shoves Previously Defined Routes Some embodiments alsopenalize expansions that shove routes of other nets along edges orwithin faces.

a. Shoving Other Routes Along Edges A potential expansion to an edgemight cause the other net routes on the edge to have to be shoved alongthat edge. FIG. 39 illustrates a process 3900 that generates a line PLFthat expresses the extra shoving cost of an expansion to an exit on anedge with one or more net routes crossing it. This process is describedby reference to FIG. 40, which illustrates an expansion from an exit4002 on a first edge 4004 to an exit 4006 on a second edge 4008. In FIG.40, a previously defined route 4010 of another net crosses the secondedge 4008. This route also crosses edges 4004 and 4028. Joints 4018,4030, and 4028 specify this route's crossing of edges 4004, 4008, and4028.

As shown in FIG. 39, the process 3900 initially determines (at 3905)whether the route of at least one other net intersects the edge thatcontains the destination exit. If not, the process terminates.Otherwise, the process 3900 selects (at 3907) one of the routes thatcrosses the destination exit's edge.

Next, at 3910, the process identifies the vectors to project from theselected route's joints on the edges that the shoved route crossesbefore and after the destination edge (i.e., before and after the edgecontaining the expansion's destination exit). Some embodiments projectvectors in each available interconnect-line direction that falls withina four-sided polygon that is formed by the selected route's two jointsand by two points that define the exit's domain (i.e., maximum possibleboundaries) along its edge. The boundary-defining points are definedbased on the spacing and width constraints and obstacle constraints onthe destination exit's edge. For instance, in FIG. 40, points 4016 and4024 are the two boundary-defining points of the exit. The destinationexit's boundaries cannot be defined beyond these two points withoutviolating required spacing and width constraints between the net routescrossing the edge and between the nets and obstacles near the edge.

The points 4016 and 4024 define a four-sided polygon along with joints4018 and 4026. Within this polygon, two vectors 4012 and 4014 areprojected from joint 4018 on edge 4004, and one vector 4020 is projectedfrom joint 4026 on edge 4028. By projecting vectors from the joints 4018and 4026, the process 3900 assumes that these joints will remain fixedas the route's joint 4030 is shoved along the edge 4008. FIG. 40 alsoillustrates a vector 4022 that emanates at a 135° angle from joint 4026.This vector is not projected as it does not fall within the polygondefined by points 4016, 4018, 4024, and 4026 (i.e., this vector wouldintersect the edge 4008 outside of the destination exit's domain).

After identifying (at 3910) the vectors to project from the previous andnext joints of the shoved route, the process 3900 identifies (at 3915)the intersection of the identified vectors and the destination edge. InFIG. 40, projection vectors 4012, 4014, and 4020 intersect thedestination edge at points 4034, 4036, and 4038.

At 3915, the process specifies a line PLF that is defined over thedestination exit's domain and that expresses the cost of shoving theroute along the destination exit's domain. The expansion to thedestination exit necessarily requires the shoving of one or more routeson the edge when this exit's edge does not have any portion to which thepath can expand without shoving the selected route. In this situation,the process specifies a knot in the line PLF for each intersection pointidentified at 3910. In this situation, the process also specifies twoknots for the two points that define the exit's maximum possibleboundaries along its edge. For each knot that the process specifies inthis situation, the process computes a PLF-value that equals (1) thedistance (according to equation (2)) between the knot's location and thelocation of the selected route's joint on the previous edge, plus (2)the distance (according to equation (2)) between the knot's location andthe location of the selected route's joint on the next edge, minus (3)the current length (according to equation (2)) of the route between itsprevious and next joints.

On the other hand, when the destination exit's edge has a segment towhich the path can expand without shoving any route on the edge, theprocess specifies a knot at each identified intersection point that isoutside the segment that can be expanded to without shoving otherroutes. In this situation, the process also specifies a knot at each ofthe two boundary points of this segment (i.e., the segment to which thepath can expand without shoving any route) and sets the PLF-value ofthese knots to zero. The PLF-value of each of the other knots that theprocess specifies at this stage equals (1) the distance (according toequation (2)) between the knot's location and the location of theselected route's joint on the previous edge, plus (2) the distance(according to equation (2)) between the knot's location and the locationof the selected route's joint on the next edge, minus (3) the currentlength (according to equation (2)) of the route between its previous andnext joints.

In the example illustrated in FIG. 40, the PLF-value of a knot specifiedat point 4038 is (1) the distance (according to equation (2)) betweenpoint 4038 and joint 4018, plus (2) the distance (according to equation(2)) between point 4038 and joint 4026, minus (3) the length (accordingto equation (2)) of the route between its joints 4018 and 4026.

The line PLF that the process defines at 3915 expresses the additionalwirelength due to shoving of the selected route along the edge. After3915, the process determines (at 3920) whether it has generated such aline PLF for each route that crosses the destination exit's edge. Ifnot, the process returns to 3907 to select another route and to repeatoperations 3910–3920 for this newly selected route. When the processdetermines (at 3920) that it has generated a line PLF for each routethat crosses the destination exit's edge, it adds (3925) the PLF's thatit generated if it generated more than one PLF. After 3925, it thenterminates.

b. Shoving Other Routes in a Face A potential expansion to a surface(e.g., a hole or a portion of a hole) within a face might cause theother net routes that intersect the face to be shoved within the face.FIG. 41 illustrates a process 4100 that generates a surface PLF thatexpresses the extra shoving cost of an expansion to a surface that hasone or more other net routes crossing its face. This process isdescribed by reference to FIG. 42, which illustrates an expansion froman exit 4202 on a first edge 4204 to a hole 4206 in a face 4208. In FIG.42, a previously defined route 4210 of another net crosses the face4208. This route crosses the first edge 4204 and a second edge 4212 attwo joints 4214 and 4216.

As shown in FIG. 41, the process 4100 initially determines (at 4105)whether the route of at least one other net intersects the face thatcontains the destination hole. If not, the process terminates.Otherwise, the process 4100 selects (at 4110) a route that intersectsthe destination surface's face. It then identifies (at 4115) twolocations from which to project vectors on the edges of the destinationsurface's face. In some embodiments, these two locations are a distanceS away from the selected route's joints along the edges of these joints.The distance S accounts for the via width and the minimum spacingbetween the current net's route (i.e., the route of the net currentlybeing routed) and the selected route. FIG. 42 illustrates two points4220 and 4222 that are a distance S away from joints 4214 and 4216 ofthe route 4210.

Next, the process 4100 projects (at 4120) vectors from the locationsidentified at 4105. Some embodiments project vectors in each availableinterconnect-line direction that falls within the destination surface'sface. FIG. 42 illustrates (1) three vectors 4224, 4226, and 4228 thatare projected from location 4220 on edge 4204, and (2) three vectors4230, 4232, and 4234 that are projected from location 4222 on edge 4212.By projecting vectors from locations 4220 and 4222 that are identifiedfrom joints 4214 and 4216, the process 4100 assumes that the joints 4214and 4216 will remain fixed as the route is shoved within face 4208.

After projecting vectors at 4120, the process 4100 identifies (at 4125)the intersection of the projected vectors (1) with the boundary of thedestination polygonal surface and (2) with each other within theboundary of the destination polygonal surface. In FIG. 42, theprojection-vector intersections are at points 4240 and 4242. Theprojection vectors intersect the boundary of the destination surface at4246–4256.

At 4130, the process specifies a surface PLF that is defined over thedestination surface and that expresses the cost of shoving the route inthis surface. The process specifies a knot at each identifiedintersection point, unless the intersection point is within a portion ofthe destination surface to which the path can expand without shoving theselected route. The process sets the PLF-value of each knot that isspecified at such an intersection point equal to (1) the distance(according to equation (2)) between the knot's location and the locationof the selected route's joint on one edge of the destination surface'sface, plus (2) the distance (according to equation (2)) between theknot's location and the location of the selected route's joint on theother edge of the destination surface's face, minus (3) the currentlength (according to equation (2)) of the route between the two joints.For instance, the PLF-value of a knot specified at point 4240 is the (1)the distance between point 4240 and location 4220, plus (2) the distancebetween point 4240 and location 4222, minus (3) the length of the routebetween joints 4214 and 4216. Similarly, the PLF-value of a knotspecified at intersection point 4246 equals the sum of the distancebetween this point and points 4220 and 4222, minus the length of theroute between joints 4214 and 4216.

The intersection of one or more projected vectors with each other orwith the destination-surface boundary might fall within a portion of thedestination surface to which a path can expand without shoving theroute. Of these intersection points, knots are specified only for thosepoints that might define the boundary between the destination-surfaceportion that a path can expand to without shoving the selected route andthe destination-surface portion that a path can only expand to byshoving the selected route. In general, the process 4100 specifies threeor more knots to define the facet for the destination-surface portionthat a path can expand to without shoving the selected route. This facetis a zero-shove cost facet, and hence the PLF-value of its knots will bezero. In FIG. 42, the points 4220, 4254, 4260, and 4266 specify theportion of the destination surface to which a path can expand withoutshoving the route 4210. In some cases, a destination surface will nothave any portion to which a path can expand without shoving the selectedroute. At 4125, the process also specifies knots at the remainingunexamined vertices of the destination surface. For instance, in theexample illustrated in FIG. 42, the process specifies knots at thevertices 4262 and 4264.

After specifying all the knots, the process specifies (at 4130) theedges and facets of the surface PLF from the specified knots and thevectors that were used to define these knots. The process also computesand specifies (at 4130) the normal and z-intercept values for eachspecified facet. After 4130, the process determines (at 4135) whether ithas generated a surface PLF for each route that crosses thedestination-hole's face. If not, the process returns to 4110 to selectanother route and to repeat operations 4115–4135 for this newly selectedroute. When the process determines (at 4135) that it has generated asurface PLF for each route that crosses the destination surface's face,it adds (4140) the PLF's that it generated if it generated more than onePLF. After 4140, it then terminates.

VI. Mathematical Operations

A. Adding PLF's

The process 1500 adds point, line, and surface PLF's. Adding two PLF'sat a point is trivial, as their values are simply added. A constantfunction can be added to a line or surface PLF by incrementing thePLF-values of all knots of the line or surface PLF by value of theconstant function.

1. Adding Two Line PLF's

Adding two convex line PLF's that are defined across the same domain canalso be efficiently done by taking advantage of the piecewise linearnature of the functions. At any point Q, the sum of two convex linePLF's PLF1 and PLF2 that are defined over the same line is the sum ofPLF-values of PLF1 and PLF2 at that point, as illustrated by theequation below.V=PLF1(Q)+PLF2(Q).

Accordingly, in some embodiments, the process 1500 adds two line PLF'sby performing the following two operations. First, it initializes tonull a PLF3 that is to represent the sum of the two PLF's. Second, itspecifies a knot in PLF3 at each unique knot location in either PLF1 orPL2. The value of each specified knot in PLF3 is the sum of thePLF-values in PLF1 and PLF2 at the specified knot's location. In someembodiments, the process 1500 examines the knots of PLF1 and PLF2 basedon the order that they appear, i.e. the process starts from one end ofthe line over which the PLF's are defined and traverses to the otherend. If a first line PLF (F1) is defined over a larger portion of a linethan a second line PLF (F1), the first and second PLF's can be summedover their overlapping portion of their domains by (1) specifying athird line PLF (F3) that is equal to the first PLF but is only definedover the overlapping portion, and (2) performing the above describedapproach to obtain the sum of the second and third PLF's (F2 and F3).

2. Adding Two Surface PLF's

FIG. 43 illustrates a specific process 4300 for adding two surface PLF'sthat have overlapping but potentially different domains. To add twosurface PLF's, this process performs a line sweep operation thatidentifies the intersection of the two polygonizations. This process isdescribed by reference to FIG. 44, which illustrates an example wheretwo surface PLF's 4405 and 4410 are added to produce a third surface PLF4415. In FIG. 44, the PLF's are projected onto the x,y plane in order tosimplify their visual presentation.

As shown in FIG. 43, the process 4300 initializes (at 4305) a new PLF(“result PLF”) that will eventually represent the sum of the two PLF's.Next, at 4310, the process identifies the leftmost knot location ineither PLF, and sets a variable CurX to the x-coordinate of theidentified knot location. The process then selects (at 4315) a knot ineither PLF that has an x-coordinate equal to CurX. There might beseveral knots with a particular x-coordinate in each PLF. To addressthis situation, some embodiments use the convention that when two ormore knots in a PLF have the same x-coordinate, the process shouldselect (at 4315) the knot that has the smallest y-coordinate and thathas not yet been examined. At 4315, the process specifies a knot in theresult PLF at the x,y location of the selected knot, if one such knothas not already been specified. At 4315, the process sets the PLF-valueof the specified knot as the sum of the PLF-values of the two PLF's atthe specified knot's x,y coordinates.

As further described below, each time the process 4300 encounters a knotthat “opens” an edge (i.e., encounters a new edge) in its left-to-rightsweep, it adds the edge to a list of open facets for the selected knot'sPLF. Some embodiments use the convention that a knot opens an edge if(1) its x-coordinate is smaller than the x-coordinate of the other knotof its edge, or (2) its y-coordinate is smaller than the y-coordinate ofthe other knot of its edge when both edge knots have the samex-coordinate.

Before adding a new edge to the open list, the process 4300 determines(at 4320) whether the selected knot closes any edge in the selectedknot's PLF. In some embodiments, a knot closes an edge if (1) itsx-coordinate is larger than the x-coordinate of the other knot of itsedge, or (2) its y-coordinate is larger than the y-coordinate of theother knot of its edge when both edge knots have the same x-coordinate.If the process determines (at 4320) that the selected knot does notclose any edges, the process transitions to 4345, which is furtherdescribed below. If the selected knot does close an edge, the processselects (at 4325) an edge that the selected knot closes. It thenperforms (at 4330) an intersection operation between the closed edge andeach edge on the open list of the other PLF.

FIG. 45 illustrates the intersection process 4500. This processinitially compares (at 4505) the closed edge with each edge on the openlist of the other PLF in order to identify each intersection of theclosed edge with another edge. It then determines (at 4510) whether itidentified at 4505 the intersection at a point of the closed edge withany of the open edges of the other PLF.

If the process determines (at 4510) that the closed edge does notintersect any open edge, the process adds (at 4515) the closed edge tothe result PLF and then terminates. Otherwise, the process specifies (at4520) a knot at each point that the closed edge intersects one of theopen edges of the other PLF. The value of each specified knot is the sumof the two PLF's at the location of the specified knot. For instance, inFIG. 44, when the edge between knots 4458 and 4460 in the first PLF 4405closes, the process 4500 identifies this edge's intersection with theedge between knots 4462 and 4464 in the second PLF 4410. At theintersection of these two edges, the process 4500 identifies a knot 4455in the third PLF 4415. The value of the knot 4455 equals the sum of thevalues of PLF's 4405 and 4410 at this knot's x, y coordinates.

After specifying one or more knots at 4520, the process 4500 defines (at4525) two or more collinear edges to replace the closed edge, and addsthe collinear edges to the result PLF. In FIG. 44, the processidentifies (at 4525) two new edges, one between knots 4450 and 4455, andone between knots 4455 and 4454. It then adds these two edges to thedefinition of the third PLF 4415, in place of adding the closed edgebetween knots 4450 and 4454.

For each particular edge on the open list of the other PLF thatintersects the closed edge, the intersection process 4500 (at 4530) (1)defines two new edges, (2) adds one of the two edges to the result PLF,and (3) replaces the particular edge with the other of the two edges.The process uses the knot specified at 4520 at the intersection of theclosed edge and an open edge to define the two new edges for the openedge. For instance, in FIG. 44, the process (at 4530) would define twoedges for the open edge between knots 4462 and 4464. One of the twoedges is between knots 4462 and 4455, and one edge is between knots 4455and 4464. The process would add the edge between knots 4455 and 4464 tothe result PLF. It would also replace the edge between knots 4462 and4464 on the open list of PLF 4410 with the edge between knots 4455 and4464. After 4530, the process terminates.

After the process 4300 performs the intersection process at 4330, theprocess 4300 removes (at 4335) the selected closed edge from its openlist. It then determines (at 4340) whether it has examined all the edgesthat the selected knot (i.e., the knot selected at 4315) closes. If not,the process transitions back to 4330 to select another closed edge.

Otherwise, the process adds (at 4345) any edge that the selected knotopens to the list of “open” edges for the selected knot's PLF. After4345, the process determines (at 4350) whether it has examined each knotin each PLF with an x-coordinate CurX. If not, the process transitionsback to 4315 to select another knot with the x-coordinate CurX. If so,the process transitions to 4355, where it determines whether it hasexamined all the unique x-coordinates of knots in either PLF. If not,the process sets (at 4360) CurX to the next leftmost x-coordinate of anyknot in either PLF.

When the process determines (at 4355) that it has examined all theunique x-coordinates of knot locations in both PLF's, it examines (at4365) the edges added to the result PLF and identifies facets formed byrelated sets of edges. At 4365, the process also identifies the normaland z-intercept of each facet based on the knot values of the facet.When one (F1) of the PLF's being summed is defined over a larger portionof a surface than the other PLF (F2), the process 4300 can limit (at4365) the surface PLF's definition that represents the sum of these twoPLF's to cover only the overlapping domain of the two PLF's.Alternatively, before 4305, the process 4300 could generate a third PLF(F3) that is equal to the first PLF but is only defined over the domainof the second PLF (i.e., only defined over the portion of the surfacethat both PLF's F1 and F2 are defined).

FIG. 44 illustrates the result of the addition of the two PLF's 4405 and4410. In this figure, the first PLF 4405 has five boundary knots4430–4438 that are at the same location as five boundary knots 4440–4448of PLF 4410. For these five pairs of knots, the process identifies fiveknots 4420–4428 in the third PLF 4415. In the third PLF 4415, theprocess also identifies four knots for the other four unique knots4458–4462 in the two PLF's 4405 and 4410. As described above, it alsospecifies a knot 4455 at the intersection of two non-boundary edges ofthe two PLF's. In turn, these knots specify 13 edges that define fourfacets 4466–4472.

B. Filtering PLF's

After identifying the G PLF of a potential expansion, thepath-generation process 1500 filters (at 1534) the expansion's G PLFwith the filter PLF of the destination particle. This filteringoperation also sets the destination particle's filter PLF equal to theminimum of its original filter PLF and the expansion's G PLF. Filteringa point PLF for an expansion to a point is trivial. The expansion isdiscarded when its PLF-value is greater than the filter function valueat that point. Otherwise, it is kept, and the point's filter function isset to the expansion's PLF-value.

1. Filtering Two Line PLF's and Identifying their Minimum

FIG. 46 illustrates a process 4600 that performs the filtering andminimum operations for an expansion to a line (i.e., an expansion to anexit or a wall). Specifically, this function filters a filtered PLF(PLF1) by a filter PLF (PLF2), which may be non-convex. In theembodiments described below, the filter PLF2 is defined across atopological particle, while the filtered PLF1 might be defined acrossonly a portion of the topological particle. Accordingly, the process4600 performs its filtering and minimum operations for only the portionof the particle over which the filtered PLF1 is defined. Otherembodiments might perform the filtering over the entire particle. In theembodiments described below, the filtering operation discards theportions of the filtered PLF that are larger than the correspondingportions of the filter PLF. Other embodiments, however, might notdiscard these portions, but rather might set the PLF-values in theseportions to infinite. These embodiments then would not need to defineseveral drops when several pieces of the filtered PLF remain after thefilter.

As shown in FIG. 46, the process 4600 initially sets (at 4605) a statevariable to “Unknown” and sets two PLF's PLF3 and PLF4 to null. Theprocess uses PLF3 to represent the minimum of the filtered and filterPLF, while it uses PLF4 to keep track of the remaining portions of thefiltered PLF. As the minimum, PLF3 will represent at each offset Q alongthe line the minimum of PLF1(Q) and PLF2(Q). PLF3 might end up being anon-convex PLF. As PLF1 and PLF2 are, PLF3 is defined over the line andwill be represented by a sequence of knots that is sorted based on theoffset values of the knots.

Next, at 4610, the process identifies the unique knot locations in bothPLF1 and PLF2 and generates a set of unique knot locations that aresorted based on the offset values of the knots in this set. The processthen selects (at 4615) the first location in the sorted set. The processthen determines (at 4620) whether the filtered PLF1 is greater than thefiltered PLF2 at the location selected at 4615.

If not, the process determines (at 4625) whether the state is PLF1_Loweror unknown. When the process determines at 4625 that the state isPLF1_Lower or is unknown, the process (at 4635) specifies a knot atselected location, sets its value to the value of PLF1, and adds thespecified knot to PLF3 and PLF4. At 4635, the process also sets thestate to PLF1_Lower if the state was previously unknown. In thissituation, the process performs these operations because (1) if thestate is unknown, this knot is at the first knot location and it is fromthe filtered PLF1, and (2) if the state is PLF1_Lower, this knot is fromthe same PLF as the last knot added to PLF3 (i.e., PLF1 and PLF2 havenot crossed between the last knot added to PLF3 and this knot, andduring this internal PLF1 has been the smaller PLF). From 4635, theprocess transitions to 4655, which will be further described below.

On the other hand, when the process determines at 4625 that the state isPLF2_Lower, the knot at selected location is from a different PLF thanthe last knot added to PLF3. Hence, PLF1 and PLF2 have crossed betweenthe last knot added to PLF3 and this knot. During this internal, PLF2started smaller and ended larger. Accordingly, the process transitionsto 4630 to perform several operations to account for this crossing. At4630, the process initially sets the state to be PLF1_Lower. It thenidentifies the offset value Q along the edge where PLF1 and PLF2intersected between the last knot added to PLF3 and the knot at thelocation selected at 4615. This intersection can be easily computed asit is an intersection between two line segments representing PLF1 andPLF2 between the last knot added to PLF3 and the knot at selectedlocation. The process then specifies a knot at intersection point Q andadds this knot to PLF3 and PLF4. The PLF-value of this added knot is thePLF-value of either PLF 1 or PLF2 at the offset-value of theintersection point Q. If the PLF-value of a knot at the selected knotlocation is smaller than the other PLF's value at this location, theprocess specifies a knot at the selected knot location, sets its valueto the value of PLF1, and adds the specified knot to the description ofPLF3 and PLF4. From 4630, the process then transitions to 4655.

If the process determines (at 4620) that the filtered PLF1 is greaterthan the filter PLF2 at the location selected at 4615, the processdetermines (at 4640) whether the state is PLF2_Lower or unknown. Whenthe process determines at 4640 that the state is PLF2_Lower or unknown,the process (at 4645) specifies a knot at selected location, sets itsvalue to the value of PLF2 at this location, and adds the specified knotto PLF3. At 4645, the process also sets the state to PLF2_Lower if thestate was previously unknown. The process performs these operations at4645 because (1) if the state is unknown, this knot is the first knotselected, and it is from the filter PLF2, and (2) if the state isPLF2_Lower, this knot is from the same PLF as the last knot added toPLF3 (i.e., the two PLF's have not crossed between the last knot addedto PLF3 and this knot, and during this interval PLF2 has been thesmaller PLF). From 4645, the process transitions to 4655.

On the other hand, when the process determines at 4640 that the state isPLF1_Lower, the knot at selected location is from a different PLF thanthe last knot added to PLF3. Hence, the two PLF's have crossed betweenthe last knot added to PLF3 and the knot selected at 4615. During thisinterval, PLF1 started smaller and ended larger. Accordingly, theprocess transitions to 4650 to perform several operations to account forthis crossing. At 4650, the process initially sets the state to bePLF2_Lower. It then identifies the offset value Q along the edge wherePLF1 and PLF2 intersected between the last knot added to PLF3 and theknot selected at 4615. This intersection can be easily computed, as itis an intersection of two line segments representing PLF1 and PLF2between the last knot added to PLF3 and the knot at selected location.The process then specifies a knot at intersection point Q and adds thisknot to PLF3 and PLF4. The PLF-value of this added knot is the PLF-valueof either PLF1 or PLF2 at the offset-value of the intersection point Q.If the PLF-value of a knot at the selected knot location is smaller thanthe other PLF's value at this location, the process specifies a knot atthe selected location, sets its value to the value of PLF2 at thislocation, and adds the specified knot to the description of PLF3 andPLF4. The process then specifies a new PLF equal to PLF4 and thenreinitializes PLF4. The new PLF represents a remaining portion of thefiltered PLF. From 4650, the process transitions to 4655.

At 4655, the process determines whether it has examined all the uniqueknot locations in the sorted set produced at 4610. If not, the processreturns to 4615 to select the next unique knot location in the sortedset and to perform the subsequent operations for this newly selectedknot location. On the other hand, when the process determines (at 4655)that it has examined all the unique knot locations in the combined set,the process uses (at 4660) PLF3 to define the new filter function. Asmentioned above, the filtered function PLF1 might not be defined overthe entire domain over which the filter function PLF2 is defined.Because of this, the process 4600 performs its minimum operations foronly the portion of the filter function's particle over which thefiltered PLF1 is defined.

When the filtered function PLF1 and the original filter function PLF2have the same domain, the process sets (at 4660) the particle's filterfunction to PLF3. On the other hand, when the filtered PLF1 is definedover a smaller portion of the filter function's particle, the processreplaces the portion of the filter function (PLF2) that is defined overthe filtered function's (PLF1's) domain with PLF3. The process leavesthe portion of PLF2 outside of PLF1's domain intact. A boundary knot ofPLF3 will be removed if it is an original knot of the filter PLF2 or ifit is on a PLF segment in PLF3 that has the same slope as the originalfilter PLF2 at the boundary knot. At 4660, the remaining portion orportions of the filtered PLF1 are the PLF's that the process 4600defines at 4650.

As mentioned above, the path-generation process 1500 filters (at 1512)the G PLF of a drop that it retrieves from the priority queue with thefilter function of the drop's particle. For this filtering operation,the path-generation process uses a process that is identical to theprocess 4600 with the exception that unlike process 4600, the filteringprocess at 1512 does not maintain PLF3 as it does not need to identifythe minimum of the filtered and filter PLF's at 1512.

2. Filtering Two Surface PLF's

FIG. 47 illustrates a process 4700 for filtering two surface PLF's thathave overlapping but potentially different domains. To filter two suchsurface PLF's, the process 4700 performs a plane sweep operation thatidentifies the intersection of the two polygonizations. This processstarts by initializing (at 4705) a Min PLF that it will use to recordthe minimum of the two surface PLF's that it is filtering.

It then identifies (at 4710) the leftmost unique knot location (i.e.,the unique knot location with the smallest x-coordinate) in either PLF,and sets a variable CurX to the x-coordinate of the identified knotlocation. The process then selects (at 4715) a knot in either PLF thathas an x-coordinate equal to CurX. There might be several knots with aparticular x-coordinate in each PLF. To address this situation, someembodiments use the convention that when two or more knots in a PLF havethe same x-coordinate, the process should select (at 4715) the knot thathas the smallest y-coordinate and that has not yet been examined.

As further described below, each time the process 4700 encounters a knotthat “opens” a facet (i.e., encounters a new facet) in its left-to-rightsweep, it adds the facet to a list of open facets for the selectedknot's PLF. To address the case where the two knots of a facet have thesame x-axis coordinate, some embodiments use the convention that a knotopens a facet if (1) it is the one with the smallest x-coordinate whenall the facet's knots have different x-coordinates, or (2) it is the onewith the smallest x-coordinate and the smallest y-coordinate when twofacet's knots have the smallest x-coordinate of all knots of the facet.

Before adding a new facet to the open list, the process 4700 determines(at 4720) whether the selected knot closes a facet in the selectedknot's PLF. In some embodiments, a knot closes a facet if (1) the knotis the one with the largest x-coordinate when all the facet's knots havedifferent x-coordinates, or (2) it is the one with the largestx-coordinate and the largest y-coordinate when two facet's knots havethe largest x-coordinate of all knots of the facet. If the processdetermines that the selected knot does not close any facet, the processtransitions to 4745, which is further described below. However, if theselected knot closes at least one facet, the process selects (at 4725) afacet that the selected knot closes. It then performs (at 4730) a minoperation between the closed facet and each facet on the open list ofthe other PLF.

FIG. 48 illustrates the process 4800 for identifying the minimum of twofacets F1 and F2 from two different PLF's. This process initiallyidentifies (at 4805) a polygon that represents the intersection of thedomains of the two facets. Next, at each vertex of the polygon, theprocess (at 4810) identifies and records the PLF that provides theminimum PLF-value. In case both PLF's provide the same value at avertex, the process records the identity of the filtered PLF for thevertex at 4810. At 4810, the process also specifies a knot at eachvertex of the intersection polygon. The PLF-value of a specified knotfor a vertex is the value of the recorded PLF at the vertex.

Next, the process determines (at 4815) whether one PLF provided all thelowest values at all the vertices of the polygon identified at 4805. Ifnot, the two facets intersect and the process performs 4820–4830 toaccount for this intersection. These operations will be described byreference to FIG. 49, which illustrates two facets 4952 and 4954 thatintersect. At 4820, the process 4800 identifies an edge along which thetwo PLF's intersect, and identifies or specifies two knots thatrepresent this edge. FIG. 49 illustrates the identified edge 4956between the two facets 4952 and 4954, and the two knots 4958 and 4960that define this edge.

At 4825, the process then (1) uses the knots identified at 4820 tospecify two new facets F3 and F4 that represent the minimum of theintersected facets F1 and F2, and (2) adds these new facets F3 and F4 tothe Min PLF. The two new facets will include the two knots identified at4820 and one or more knots that were specified (at 4810) at theintersection polygon vertices. Each new facet will have edges betweenits knots.

FIG. 49 illustrates the two new facets 4962 and 4964. Facet 4962 is theportion of facet 4954 that is below than the corresponding portion ofthe facet 4952. Similarly, facet 4964 is the portion of facet 4952 thatis below the corresponding portion of the facet 4954. The facet 4962 isrepresented by edges between knots 4958, 4960, 4966, and 4968, while thefacet 4964 is represented by edges between knots 4958, 4960, 4970, and4972. The process can obtain the normal and z-intercept attributes ofeach specified facet 4962 and 4964 from the open or closed facet 4954and 4952 that corresponds to the specified facet. Alternatively, theprocess can compute the normal and z-intercept of each facet that itspecifies at 4825 from its knots.

Next, at 4830, the process adds the facet that it specified at 4825 forthe smaller portion of the filtered PLF to a list that stores allremaining facets of the filtered PLF. If the facet 4954 in FIG. 49 isfrom the filtered PLF, the process would add (at 4830) the facet 4962 tothe list of the remaining facets of the filtered PLF.

If the process 4800 determines (at 4815) that one PLF provided thelowest values at all vertices of the polygon identified at 4805, theprocess specifies (at 4835) a new facet that is defined over the polygonand that is identical to the PLF over this polygon. It sets the valuesat these knots and edges from the smaller PLF, and it sets the specifiedfacet's normal and z-intercept identical to the facet of the smallerPLF. The process then writes (at 4840) the facet specified at 4835 andits edges and knots into the Min PLF. If the smaller PLF is the filteredPLF, the process also adds (at 4845) this information (i.e., the facetspecified at 4835 and its edges and knots) to the facet list that storesthe remaining facets of the filtered PLF. After 4830 and 4845, theprocess 4800 terminates.

After the process 4700 performs (at 4735) the min process 4730 for eachcombination of the closed facet and the open facets on the other PLF'slist, the process 4700 removes (at 4735) the selected closed facet fromits open list. It then determines (at 4740) whether it has examined allthe facets that the selected knot (i.e., the knot selected at 4715)closes. If not, the process transitions back to 4725 to select anotherclosed facet. Otherwise, the process adds (at 4745) any facet that theselected knot opens to the list of “open” facets for the selected knot'sPLF. The process then determines (at 4750) whether it has examined eachknot in each PLF with an x-coordinate CurX. If not, the processtransitions back to 4715 to select another knot with the x-coordinateCurX. If so, the process transitions to 4755, where it determineswhether it has examined all the unique x-coordinates of knots in eitherPLF. If not, the process sets (at 4760) CurX to the next leftmostx-coordinate of any knot in either PLF.

When the process determines (at 4755) that it has examined all theunique x-coordinates of knot locations in either PLF, it examines (4765)the facets added to the filtered list, and identifies connected sets offacet. Each connected sets of facets represents a remaining portion ofthe filtered PLF. It associated each connected sets of facet so that theset represents one remaining portion of the filtered PLF. It alsoremoves any unnecessary edges and knots (if any) from each associatedset and modifies the definition of one or more facets of the setaccordingly.

At 4765, the process also defines the filter PLF as the Min PLF. ThisMin PLF might include unnecessary edges and knots (e.g., at theboundaries of the facets that the process 4800 intersected). If so,before setting the filter PLF to the Min PLF, the process removes theseunnecessary edges and knots at 4765 and modifies the definition of oneor more facets of the Min PLF accordingly.

As mentioned above, the filter PLF might be defined over a larger domainthan the filtered PLF. As an alternative to the above-describedapproach, some embodiment might perform the process 4700 for only theportion of the surface over which the filtered PLF1 is defined. In theseembodiments, the filter PLF would be defined over a larger domain thanthe Min PLF. Accordingly, for such cases, the process 4700 only sets theportion of the filter PLF that is defined over the same domain as thefiltered PLF to the Min PLF. The process retains the portions of thefilter PLF that falls outside of the domain of the filtered PLF. Toaccount for the boundary between retained and replaced portions of thefilter PLF, the process might need to add knots and edges to the filterPLF's description and might need to modify the filter PLF's descriptionof facets at the boundaries.

As mentioned above, the path-generation process 1500 filters (at 1512)the G PLF of a drop that it retrieves from the priority queue with thefilter function of the drop's particle. For this filtering operation,the path-generation process uses a process that is identical to theprocess 4700 with the exception that unlike process 4700, the filteringprocess at 1512 does not maintain Min as it does not need to identifythe minimum of the filtered and filter PLF's at 1512.

VII. Computing the Ĥ PLF

In some embodiments, the process 1500 specifies the Ĥ PLF of a particleas the lower-bound distance between the particle and a bounding octagonthat encompasses the set of target particles for a path search. Someembodiments identify such a lower-bound distance by (1) identifying anaxis-aligned rectangular box (i.e., a box that is aligned with the x-ycoordinate axes of the layout) and a 45° rotated rectangular box (i.e.,a box that is aligned with an s-t coordinate axes that is rotated by 45°with respect to the layout's x-y axes) that enclose the set of targetparticles, (2) identifying two PLF's that express the distance betweenthe particle and each of these boxes, and then (3) defining the H PLF asthe maximum of the two identified PLF's.

A. Identifying the Bounding Boxes

In some embodiments, the process identifies an axis-aligned bounding boxby (1) identifying the minimum and maximum x- and y-coordinates(X_(MIN), X_(MAX), Y_(MIN), and Y_(MAX)) of the set of points in thedesign layout's coordinate space, and (2) specifying the four verticesof the box as (X_(MIN), Y_(MIN)), (X_(MIN), Y_(MAX)), (X_(MAX),Y_(MIN)), and (X_(MAX), Y_(MAX)). In some embodiments, the process 1500identifies the minimum and maximum x- and y-coordinates of the set ofpoints by examining the x- and y-coordinates of vertex points in the setof target particles. Each standalone point in this set is a vertexpoint. Also, each point in the target set that is a vertex of apolygonal shape in the set is a vertex point of the set.

The rotated bounding box is defined with respect to a coordinate systemthat is rotated by 45° with respect to the layout's coordinate system.The rotated coordinate system includes an s-axis that is at 45° withrespect to the x-axis of the layout's coordinate space, and a t-axisthat is at 135° with respect to the x-axis of the layout's coordinatespace. To identify the 45° rotated bounding box, the process 1500 insome embodiments first maps the coordinates of the vertex points of theset of target particles from the layout's coordinate system to the 45°rotated coordinate system. The process performs this mapping by usingthe equation below.

${S_{1} = \frac{X_{i} - Y_{i}}{\sqrt{2}}},{{{and}\mspace{14mu} T_{1}} = {\frac{X_{i} + Y_{i}}{\sqrt{2}}.}}$In this equation, (1) X₁ and Y_(i) are the coordinates in the layout'sx- and y-axes for the i-th vertex point in the target set, and (2) S₁and T_(i) are the coordinates in the rotated s- and t-axes for the i-thvertex point in the target set. If the rotated bounding box is rotatedcounterclockwise at an arbitrary angle θ with respect to the layout'scoordinate system, the following equation could be used to perform themapping.

$\begin{pmatrix}S_{i} \\T_{i}\end{pmatrix} = {{\begin{matrix}{\cos\;\theta} & {{- \sin}\;\theta} \\{\sin\;\theta} & {\cos\;\theta}\end{matrix}}\;{\begin{matrix}X_{i} \\Y_{i}\end{matrix}}}$

After performing the mapping, the process 1500 then identifies a 45°bounding box that is defined with respect to the 45° rotated coordinatesystem and that surrounds the set of routed points in the s- and t-axes.In some embodiments, the process (1) identifies the minimum and maximums- and t-coordinates (S_(MIN), S_(MAX), T_(MIN), and T_(MAX)) of eachvertex point in the set of routed points, and (2) specifies the fourvertices of the rotated box as (S_(MIN), T_(MIN)), (S_(MIN), T_(MAX)),(S_(MAX), T_(MIN)), and (S_(MAX), T_(MAX)).

B. Computing the PLF's

In some embodiments, the process 1500 computes a PLF that represents thedistance between a particle and a bounding box in an analogous manner tothe propagating of a zero-cost function that is defined over thebounding box to the particle. FIG. 50 illustrates this computation forsome embodiments that use the wiring model that specifies horizontal,vertical, and ±45° diagonal interconnect lines on each layer.Specifically, this figure illustrates an axis-aligned bounding box 5005and a 45° rotated bounding box 5010. Three vectors are projected fromeach vertex of each bounding box. FIG. 50 illustrates the three vectorsthat are projected from each vertex of the boxes 5005 and 5010. Someembodiments do not project vectors in all interconnect directions.Instead, they project only the propagation vectors that will intersectthe destination particle. These propagation vectors are the vectors thatfall within a triangle defined by the bounding box's vertex point andthe leftmost and rightmost vertices of the destination particle.

If the topological particle is a point, the H PLF is a single value thatis defined for the point. This value is the distance between the pointand the bounding box. This distance is computed by (1) identifying thewedge or channel that contains the point, and (2) computing the distanceaccording to the wedge and channel distance rules described above. Forinstance, in FIG. 50, the H PLF-value of the point 5015, which fallswithin a wedge defined by vectors 5020 and 5022 that emanate from vertex5024, is the distance between the point 5015 and the vertex 5024. On theother hand, if the particle is point 5036 that falls within the channeldefined by vectors 5026 and 5028 that emanate from the rotated box 5010,H PLF-value is the length D of a line 5038 that is parallel to vectors5026 and 5028 and that goes from the point 5036 to the edge 5032.

If the topological particle is a line, the H PLF is a line PLF that isdefined over the line. This line PLF will have knots at theintersections of the projected vectors and the line. The H PLF-value ofeach knot is simply the straightline distance between the intersectionpoint and the vertex from which the intersecting vector projected. Thisstraightline distance is along the direction of the projectedintersecting vector. Also, a knot is specified at each unexaminedendpoint of the line, and the value of such a knot depends on whetherthe endpoint falls within a channel or a wedge.

For instance, when the particle is line 5040, two knots are specified atthe intersections 5042 and 5044 of the projection vectors 5046 and 5048with the line. The PLF-value of the knot at location 5042 is thedistance D1 to the vertex 5050 along the direction of vector 5046, whilethe PLF-value of the knot at location 5044 is the distance D2 to thevertex 5050 along the direction of vector 5048. Two knots are alsospecified at the endpoints 5052 and 5054 of the line 5050. The endpointat 5052 falls within a wedge. Hence, the PLF-value of the knot specifiedat 5052 is based on the distance between point 5052 and vertex 5050. Theendpoint at 5054, however, falls in a channel, and therefore thePLF-value of the knot defined at this point is the length D3 of the 135°line segment 5056 between the point 5054 and edge 5058.

If the topological particle is a surface, the Ĥ PLF is a surface PLFthat is defined over the surface. This surface PLF will have knots atthe intersections of the projected vectors and the boundary of thesurface. It will also have knots at each unexamined vertex of thesurface. It will further have edges between the knots, and one or morefacets defined by the edges. A normal and z-intercept will also bespecified for each facet. The values of each facet's normal andz-intercept will be determined by the PLF-values of the knots thatdefine each facet.

The PLF-values of each knot that is defined at an intersection of aprojected vector is simply the straightline distance between theintersection point and the vertex from which the intersecting vectorprojected. This straightline distance is along the direction of theprojected intersecting vector. Also, the PLF-value of each knot that isspecified at an unexamined vertex of the surface depends on whether thevertex falls within a channel or a wedge.

For instance, when the particle is a surface 5060 and the bounding boxis box 5005, four knots are specified at the intersection 5062, 5064,5066, and 5068 of the projection vectors 5070 and 5070 with the surface.The PLF-value of each of these knots is the distance between the knot'slocation and the vertex that was used to define the knot. For instance,the PLF-value of the knot at location 5062 is the distance D5 betweenthe vertex 5074 and point 5064 along the direction of vector 5070. Inthis example, five other knots are specified at the unexamined vertices5076, 5078, 5080, 5082, and 5084 of the surface. Of these vertices, onlyvertex 5080 falls within a channel. Accordingly, the PLF-value of theknot specified at the vertex 5080 is the length of the line segment 5086that is parallel to vectors 5070 and 5072. The PLF-value of the knotdefined at each of the other four unexamined vertices is the distancebetween the knot's location and the bounding box vertex that projectsthe vector that defines the wedge containing the knot location. Forinstance, the PLF-value of the knot at 5082 is the distance betweenvertex 5082 and the vertex 5024 of the box 5005.

The H PLF that is defined over the surface 5060 will have eleven edges.Nine of these edges will be defined about the boundary of the surface.Two of the edges 5088 and 5090 will be defined across the surface. Thesetwo edges in conjunction with the other nine boundary edges define threefacets. The normal and z-intercept of each facet can be derived from thePLF-values of the knots that define the facet, by using standardplane-defining techniques.

C. Identifying the Maximum

After defining the Ĥ PLF's for the two bounding boxes, some embodimentsidentify the maximum of the two PLF's. Obtaining the maximum of twoPLF's is analogous to obtaining the minimum of two PLF's, except thatthe portions that are discarded are the portions with the smallerPLF-values. Accordingly, the processes for computing the maximum of twoline PLF's or two surface PLF's are analogous to the above-describedprocesses for computing the minimum of such PLF's, except that theprocesses for computing the maximum retain the line segments or surfacepieces that have the larger PLF-values.

D. Different Computation of the Ĥ PLF

Some embodiments specify the Ĥ PLF of a particle slightly differently.These embodiments first identify a bounding polygon by identifying andintersecting two bounding rectangular boxes that enclose the set oftarget particles. In some of these embodiments, one box is aligned withthe x-y coordinate axes of the layout while the other box is alignedwith an s-t coordinate axes that is 45° rotated with respect to thelayout's x-y axes. FIG. 51 illustrates an example of an octilinearbounding polygon 5125 that is identified by intersecting an axis-alignedbox 5130 and a 45° rotated box 5135.

After identifying such an octilinear bounding polygon, these embodimentsidentify a Ĥ PLF that expresses the distance between the particle andthe octilinear bounding polygon. The identification of this PLF isanalogous to propagating a zero-cost function that is defined over theoctilinear bounding polygon to the particle. Specifically, one or morevectors are projected from each vertex of the octilinear boundingpolygon. In FIG. 51, four vectors are projected from vertex 5120, threevectors are projected from each of the vertices 5110 and 5115, and twovectors are projected from vertex 5105. These vectors are identifiedbased on the following approach. For each vertex, two directions areidentified that are perpendicular to the edges incident upon the vertexand that point away from the polygon, that is, they points left from anincoming clockwise edge, or right from an incoming counter-clockwiseedge. Vectors are then projected from each vertex (1) in the twodirections identified for the vertex, and (2) in any other directionthat falls between the two identified directions. Some embodiments donot project vectors in all interconnect directions. Instead, theyproject only the propagation vectors that will intersect the destinationparticle. These propagation vectors are the vectors that fall within atriangle defined by the bounding polygon's vertex point and the leftmostand rightmost vertices of the destination particle.

After projecting the vectors from the vertices of the octilinearbounding polygon, the process then identifies a point PLF, a line PLF,or a surface PLF in the exact same manner as described above for thecase of the axis-aligned and 45°-rotated bounding boxes. In other words,the same wedge and channel approach that was described above for thebounding boxes, can be used to identify the PLF's for the octilinearbounding polygons. One of ordinary skill will realize that theabove-described H computation techniques (e.g., the wedge and channeldistance computation technique) can be used with any type of boundingpolygon, such as a convex hull or approximate convex hull. VIII. THECOMPUTER SYSTEM

FIG. 52 presents a computer system with which one embodiment of thepresent invention is implemented. Computer system 5200 includes a bus5205, a processor 5210, a system memory 5215, a read-only memory 5220, apermanent storage device 5225, input devices 5230, and output devices5235.

The bus 5205 collectively represents all system, peripheral, and chipsetbuses that communicatively connect the numerous internal devices of thecomputer system 5200. For instance, the bus 5205 communicativelyconnects the processor 5210 with the read-only memory 5220, the systemmemory 5215, and the permanent storage device 5225.

From these various memory units, the processor 5210 retrievesinstructions to execute and data to process in order to execute theprocesses of the invention. The read-only-memory (ROM) 5220 storesstatic data and instructions that are needed by the processor 5210 andother modules of the computer system. The permanent storage device 5225,on the other hand, is read-and-write memory device. This device is anon-volatile memory unit that stores instruction and data even when thecomputer system 5200 is off. Some embodiments of the invention use amass-storage device (such as a magnetic or optical disk and itscorresponding disk drive) as the permanent storage device 5225. Otherembodiments use a removable storage device (such as a floppy disk orzip® disk, and its corresponding disk drive) as the permanent storagedevice.

Like the permanent storage device 5225, the system memory 5215 is aread-and-write memory device. However, unlike storage device 5225, thesystem memory is a volatile read-and-write memory, such as a randomaccess memory. The system memory stores some of the instructions anddata that the processor needs at runtime. In some embodiments, theinvention's processes are stored in the system memory 5215, thepermanent storage device 5225, and/or the read-only memory 5220.

The bus 5205 also connects to the input and output devices 5230 and5235. The input devices enable the user to communicate information andselect commands to the computer system. The input devices 5230 includealphanumeric keyboards and cursor-controllers. The output devices 5235display images generated by the computer system. For instance, thesedevices display IC design layouts. The output devices include printersand display devices, such as cathode ray tubes (CRT) or liquid crystaldisplays (LCD).

Finally, as shown in FIG. 52, bus 5205 also couples computer 5200 to anetwork 5265 through a network adapter (not shown). In this manner, thecomputer can be a part of a network of computers (such as a local areanetwork (“LAN”), a wide area network (“WAN”), or an Intranet) or anetwork of networks (such as the Internet). Any or all of the componentsof computer system 5200 may be used in conjunction with the invention.However, one of ordinary skill in the art would appreciate that anyother system configuration may also be used in conjunction with thepresent invention.

While the invention has been described with reference to numerousspecific details, one of ordinary skill in the art will recognize thatthe invention can be embodied in other specific forms without departingfrom the spirit of the invention. For instance, the above-describedembodiments propagate cost functions to one-dimensional lines andtwo-dimensional surfaces. Other embodiments might propagate costfunctions to and from (1) a set of associated collinear points (i.e., aset of points that can be connect by a single line passing through thepoints) that form a one-dimensional state and/or (2) a set of associatedco-planar points (i.e., a set of points that fall within a twodimensional surface) that form a two-dimensional state.

Some embodiments propagate functions to a set of associated points byexamining each point independently. Specifically, for each point in theassociated set, some embodiments compute a PLF-value by identifying thewedge and/or channel containing the point, and then computing thePLF-value according to the wedge and/or channel rules described abovefor 3415 of FIG. 34.

Some embodiments propagate functions from a set of associated points by(1) using the above-described approach of FIGS. 26–31 to identify wedgeand channels vectors that emanate from the set of points, and (2)identifying the destination state's cost function based on theidentified channels and wedges. Specifically, the location of each wedgeis identified by identifying the point that is the best point in the setfor the wedge. Also, a channel vector might emanate from a point in theset that is on a line segment connecting two points from which twoadjacent wedges emanate (i.e., from which two wedges with parallelvectors emanate). Such channel vectors can be identified based on theapproach described above by reference to FIGS. 29 and 30. Afteridentifying the wedge and channel vector emanations from the set ofpoints, the destination state's cost function can be specified accordingto the approaches described above for destination points, lines,surfaces, and set of points. Thus, one of ordinary skill in the artwould understand that the invention is not to be limited by theforegoing illustrative details, but rather is to be defined by theappended claims.

1. A method of identifying a path between a set of source states and aset of target states in a space with more than two dimensions, themethod comprising: specifying at least one path that starts from onestate; after specifying the at least one path, iteratively specifyingnew paths by expanding previously specified paths to other states in thespace until a path is identified that connects a source and targetstate, wherein expanding previously specified paths includes expandingpaths in more than two dimensions of the space, wherein at least some ofthe states are non-zero dimensional states; and for at least onespecified path that includes an expansion to a particular non-zerodimensional state, computing a particular cost function that is definedover the particular non-zero dimensional state, wherein the particularcost function comprises at least two different values associated with atleast two different locations on the particular non-zero dimensionalstate.
 2. The method of claim 1, wherein expanding paths in more thantwo dimensions comprises: expanding paths in one or more planes parallelwith two particular dimensions of the space; and expanding paths in adirection perpendicular to the two particular dimensions of the space.3. The method of claim 2, wherein the space has multiple layers, whereineach layer is a plane that is parallel to the two particular dimensionsof the space, wherein an expansion that is in a direction perpendicularto the two particular dimensions of the space is an expansion to a statein the space that belongs to two different layers of the space.
 4. Themethod of claim 3, wherein the space is a graph, and the states in thegraph include points, lines, and surfaces.
 5. The method of claim 4,wherein two-dimensional surfaces are the states that belong to more thanone layer in the graph.
 6. The method of claim 5, wherein each surfacerepresents the location of a via between two different layers in thegraph.
 7. The method of claim 4 further comprising: for each specifiedpath that results from expanding a previously specified path to anotherstate in the graph, propagating a cost function that expresses the costof the previously specified path to the other state.
 8. The method ofclaim 4 further comprising: for each specified path that results fromexpanding a previously specified path to another state in the graph,computing a cost function that expresses the cost of expanding thepreviously specified path to the other state.
 9. The method of claim 8,wherein the cost function of a path that reaches a particular state isdefined over the particular state.
 10. The method of claim 8, whereinthe state that each particular path reaches is the particular path'sdestination state, wherein iteratively specifying new paths comprises ateach iteration: selecting a previously specified path that has a bestcost-function value; and only if (i) the selected path still is a bestpath for at least a portion of the selected path's destination state and(ii) the selected path's cost function over the portion still provides abest cost of all the previously-specified paths, then specifying a newset of paths by identifying expansions of the selected path to otherdestination states in the graph.
 11. The method of claim 10 furthercomprising: after identifying and computing the cost of each specifiedpath, storing an identifier for the expansion in a storage structure,wherein the identifiers are stored in the storage structure in a sortedorder based on the best cost-function value of the corresponding pathsof the identifiers, wherein selecting a previously identified path thathas the best cost-function value comprises selecting from the storagestructure the identifier that is first in the sorted order.
 12. Themethod of claim 11, wherein the best cost-function value is the lowestcost-function value.
 13. The method of claim 10 further comprising: todetermine whether a selected path is still the best path to at least aportion of the destination state of the path, determining whether thecost function for the selected path specifies as good a cost for anyportion of the selected path's destination state as a cost function thatexpresses the best cost of all paths that have reached the destinationstate.
 14. The method of claim 10, wherein, at each iteration,identifying expansions of the selected path to other destination statescomprises: identifying a set of potential expansions about the selectedpath's destination state, wherein each potential expansion specifies apotential destination state for expanding the selected path from theselected path's destination state; for each particular potentialexpansion to a potential destination state, determining whether apotential path that is obtained by extending the selected path to thepotential destination state is the best path to at least a portion ofthe potential destination state; if the potential path provides the bestexpansion to at least a portion of the potential destination state,specifying the potential path as a new path.
 15. The method of claim 14,wherein determining whether a potential path is the best path to atleast a portion of a potential destination state comprises: computing acost function that expresses a cost of the selected path to reach thepotential destination state; and determining whether the cost functionspecifies a best cost for any portion of the potential destinationstate.
 16. The method of claim 15, wherein the best cost for any portionof the potential destination state is a cost that is as good as a costproduced by a cost function that expresses the best cost of all pathsthat have reached the potential destination state.
 17. The method ofclaim 16, wherein specifying at least one path that starts from onestate comprises specifying a plurality of paths that start from aplurality of states.
 18. The method of claim 1, wherein specifying atleast one path that starts from one state comprises specifying aplurality of paths that start from one state.
 19. A method ofidentifying a path between a set of source states and a set of targetstates in a space with more than two dimensions, the method comprising:specifying at least one path that starts from one state; afterspecifying the at least one path, iteratively specifying new paths byexpanding previously specified paths to other states in the space untila path is identified that connects a source and target state, whereinexpanding previously specified paths includes expanding paths in morethan two dimensions of the space, wherein at least some of the statesare non-zero dimensional states; and for at least one specified paththat includes an expansion to a particular non-zero dimensional state,computing a particular cost function that is defined over the particularnon-zero dimensional state, wherein the particular cost function is apiecewise linear function.
 20. The method of claim 19, wherein theparticular cost function is a convex piecewise linear function.
 21. Acomputer readable medium storing a computer program for identifying apath between a set of source states and a set of target states in aspace with more than two dimensions, the computer program comprisinginstructions for: specifying at least one path that starts from onestate; after specifying the at least one path, iteratively specifyingnew paths by expanding previously specified paths to other states in thespace until a path is identified that connects a source and targetstate, wherein the instructions for expanding previously specified pathsincludes instructions for expanding paths in more than two dimensions ofthe space, wherein at least some of the states are non-zero dimensionalstates; and for at least one specified path that includes an expansionto a particular non-zero dimensional state, computing a particular costfunction that is defined over the particular non-zero dimensional state,wherein the particular cost function comprises at least two differentvalues associated with at least two different locations on theparticular non-zero dimensional state.
 22. The computer readable mediumof claim 21, wherein the instructions for expanding paths in more thantwo dimensions comprises instructions for: expanding paths in one ormore planes parallel with two particular dimensions of the space; andexpanding paths in a direction perpendicular to the two particulardimensions of the space.
 23. The computer readable medium of claim 22,wherein the space has multiple layers, wherein each layer is a planethat is parallel to the two particular dimensions of the space, whereinan expansion that is in a direction perpendicular to the two particulardimensions of the space is an expansion to a state in the space thatbelongs to two different layers of the space.
 24. The computer readablemedium of claim 23, wherein the space is a graph, and the states in thegraph include points, lines, and surfaces.
 25. The computer readablemedium of claim 24, wherein two-dimensional surfaces are the states thatbelong to more than one layer in the graph.
 26. The computer readablemedium of claim 25, wherein each surface represents the location of avia between two different layers in the graph.
 27. A computer readablemedium storing a computer program for identifying a path between a setof source states and a set of target states in a space with more thantwo dimensions, the computer program comprising instructions for:specifying at least one path that starts from one state; afterspecifying the at least one path, iteratively specifying new paths byexpanding previously specified paths to other states in the space untila path is identified that connects a source and target state, whereinthe instructions for expanding previously specified paths includesinstructions for expanding paths in more than two dimensions of thespace, wherein at least some of the states are non-zero dimensionalstates; and for at least one specified path that includes an expansionto a particular non-zero dimensional state, computing a particular costfunction that is defined over the particular non-zero dimensional state,wherein the particular cost function is a piecewise linear function. 28.The computer readable medium of claim 27, wherein the particular costfunction is a convex piecewise linear function.